A MANUAL OF
QUATERNIONS
BY
CHARLES JASPER JOLT
M.A. D.SC. F.RS. SEC.KLA.
FELLOW OF TRINITY COLLEGE, DUBLIN
ANDRKW&* PROFESSOR OF AHTRONOMY IN THE UNIVERSITY OF DUBLIN
AND ROYAL ASTRONOMER OF IRKLVND
MACMILLAN AND CO., LIMITED
NE\V YORK : THE MACMILLAN COMPANY
1905
PEEFACE.
READERS of the Life of Sir William Rowan Hamilton will
recollect that he undertook the publication of a book on
quaternions to serve as an introduction to his great volume
of Lectures. This Manual of Quaternions was intended to
occupy about 400 pages, but while the printing slowly pro-
gressed it grew to such a size that it came to be regarded
by its author as a " book of reference " rather than as a
text-book, and the title was accordingly changed to The
Elements of Quaternions. By a curious series of events
one of Hamilton's successors at the Observatory of Trinity
College has felt himself obliged to endeavour to carry out to
the best of his ability Hamilton's original intention. And on
the centenary of Hamilton's birth a Manual of Quaternions is
offered to the mathematical world.
Last year I was called upon by the Board of Trinity College
to assist in the examination for Fellowship. I had long ago
recognized that another work on quaternions was required,
and this want was forcibly brought home to me by my new
duties. A mathematician, whose time is limited, is frightened
at the magnitude of Hamilton's bulky tomes, although a closer
acquaintance with the Elements would reveal the admirable
lucidity and the logical completeness of that wonderful book,
and although the Lectures have a charm all their own. The
student wants to attain, by the shortest and simplest route, to
a working knowledge of the calculus; he cannot be expected
to undertake the study of quaternions in the hope of being
rewarded by the beauty of the ideas and by the elegance of
the analysis. And for his sake, though with reluctance I
must confess, I have abandoned Hamilton's methods of
establishing the laws of quaternions.
vi PEEFACE.
By a brilliant flash of genius Hamilton extended to vectors
Euclid's conception of ratio, A quaternion is the mutual
relation of two directed magnitudes with respect to quantity
and direction as a ratio is the mutual relation of two
undirected magnitudes with respect to - : -'; '''.'. From this
enlarged view of a ratio, the calculus of quaternions is deve-
loped in the Elements. But the way is long and winding,
and after much labour, I found I could not greatly shorten it or
make it much less indirect. I therefore adopted another plan.
The two cardinal functions of two vectors are Sa/3 and
Va/3. These functions may be defined by the statements
that Sa/3 is the product of the length of one vector into the
projection of the other upon it, and that Va/3 is the vector
which is perpendicular to a and to /3, and which contains
as many units of length as there are units of area in the
parallelogram determined by a and /3. Both these functions
enjoy some of the properties of an algebraic product. They
are distributive with respect to each of the vectors.
The product of the vector a into /3 may be defined to be
the sum of these functions,
a/3 = Sa/3 + Va/3.
This is a quaternion the sum of a scalar and a vector. A
product of a pair of vectors is distributive but not commuta-
tive. It is now necessary to define the product of a quaternion
{q) into a vector (y), and we say that it is the sum of the
product of the scalar (Sq) into y and the product of the
vector (Vg) into y, or that
From these principles it follows almost immediately that quater-
nion multiplication is associative as well as distributive.
Division is seen to be deducible from multiplication, and
on p. 12 we arrive at the important result that every function
of quaternions formed by ordinary algebraic processes is a
quaternion, scalars and vectors being considered to be special
cases.
What we may call the grammar of the subject may be said
to terminate on p. 20, the laws of combination of quaternions
having been established, the five special symbols S, V, K, T and U
PKEFACE. vii
having been defined and their chief properties explained, various
constructions for products and quotients having been made, and
the non-commutative property of multiplication having been
illustrated by conical rotations and otherwise.
In the succeeding chapters, I have not scrupled to introduce,
either in the articles in small type or in the worked examples
in small type, illustrations of the applications of quaternions
to subjects that can hardly be supposed to be familiar to the
beginner in mathematics. It is suggested in the table of con-
tents that these more difficult portions should be omitted by
a beginner at first reading. The book is, however, primarily
intended for those who commence the study of quaternions
with a fair knowledge of other branches of mathematics; in
other words, it is written for the majority of those at present
likely to read quaternions because, as yet, the subject is not
generally taught in elementary classes. On the other hand,
I have abstained from printing examples of an artificial nature,
and I have avoided unnecessary difficulties.
Although this book may be regarded as introductory to the
works of Hamilton, it may also to some extent be considered
as supplementing them. Many of the results contained in it
have appeared only in the publications of learned societies,
and many others are believed to be novel. It is possible,
therefore, that this volume may be found to have some points
of interest for the advanced student of quaternions. He will
find, for example, that quaternions lend themselves to the
treatment of protective geometry quite as readily as to investi-
gations in mathematical physics and in metrical geometry.
By means of a somewhat elaborate table of contents, modelled
on those prefixed by Hamilton to his Lectures and Elements,
and by the aid of a full index and numerous cross references,
I trust that the contents of this book will be found to be fairly
accessible to the casual reader as well as to the systematic
student. It must be remembered, however, that the objects of
a work of this nature are to introduce a subject of the highest
educational value, and to develop a powerful and comprehen-
sive calculus. Such ends can be attained only by illustration
and by suggestion, and it is not easy to tabulate methods of
investigation.
via PREFACE.
It would be impossible to overestimate what I owe to
Hamilton's Lectures on Quaternions (Dublin, 1853) and to
his Elements of Quaternions (London, 1866, 2nd edition, in
two volumes, with notes and appendices by 0. J. Joly, London,
1899, 1901). The admirable Elementary Treatise on Qua-
ternions (3rd edition, Cambridge, 1890), by the late Professor
P, G. Tait who has done so much for quaternions by his-
classical applications of Hamilton's operator V has also been
very useful. Other writers to whom I am indebted are referred
to in the text.* I am glad to have this opportunity of offering
my thanks to my respected friend, Benjamin Williamson,,
Esq., F.K.S., Senior Fellow of Trinity College, Dublin, for his-
great kindness in assisting me with a considerable portion of
the proofs. I am also indebted to him for the uninterrupted
encouragement he has given me, alike privately and in his
official capacity as a member of the governing body of Trinity
College, in my attempts to render Hamilton's work more
widely known.
CHARLES JASPER JOLY.
THE OBSERVATORY,
DUNSINK, Co. DUBLIN, 1st Jan., 1905.
*The Bibliography by Dr. Macfarlane, published by the International Association
for the promotion of the Study of Quaternions and Allied Systems of Mathematics.
(Dublin, 1904), renders unnecessary any detailed list of works on quaternions.
CONTENTS.
CHAPTER I.
THE ADDITION AND SUBTRACTION OF VECTORS.
RT. PAGE
1. Definition of a vector. ........ 1
2. Sum of two vectors. Commutative property. .... 1
3. Addition of vectors is associative and commutative. 2
4. Null sums of vectors. ......... 2
5. Use of the sign -. ......... 2
6. Multiplication of vectors by scalar coefficients. ... - 3
7. Unit of length. Tensor and versor of a vector, - 4
a=Ta.Ua.
8. Resolution of a vector along three given vectors. ... 4
CHAPTER II.
MULTIPLICATION AND DIVISION OF VECTORS
AND QUATERNIONS.
9. Definition of the scalar function S/2 of two vectors. - 6
The doubly distributive property,
10. Definition of Va/3. The doubly distributive property,
V2a2/3=22Va/3,
11. The product of two vectors defined to be ---,
a/3=Sa/3
The doubly distributive property,
2a2/3=
The relations, a/? + /3a =2Sa/3, a/3 - /3a =* 2Va/2.
x CONTENTS.
ART. PAGE
12. A quaternion is defined to be the sum of a scalar and a vector, - 9
and is expressed as a product of a pair of vectors,
? = a/3.
Addition of quaternions.
13. The product of a quaternion and a vector is defined to be dis- 9
tributive with respect to the scalar and vector of the
quaternion, or
qy=(Sq+Vq)y = $q - J+Vq. J I
and products of vectors are shown to be distributive and to
be interpretable by formula (B).
14. Laws of three mutually rectanglar unit vectors, i t j, , deduced 10
from formula (B). Multiplication of /, j, k is associative.
Hamilton's formula,
>=/W/== z y=:-i ......................... (A)
15. Multiplication of quaternions is associative. - - - - 11
16. Division reduced to multiplication. Reciprocal of a vector, - 11
-i-i- _2_
a ~a~"~Ta 3 '
a -i if Q=a.
Every function of quaternions is a quaternion.
17. The conjugate, tensor, versor, angle, plane and axis of a qua- 12
ternion.
Example giving functions for the quadrinomial form,
q = w + ix -*rjy + Jcz.
18. A quaternion expressed as a quotient of vectors, - - - 13
OB
? = ~, Lq=LA.o*.
The ratio of two vectors, Hamilton's extension of Euclid's con-
* ception,-
19. A quaternion as an operator. Effect of a vector, of a versor. - 14
20. Construction for the product of two quaternions. - - * 14
"The relation
CONTENTS.
ART. PAGE
21. Spherical representation of multiplication. Vector-arcs. - - J6
Commutative quaternions are coplanar. Square of right versor
is negative unity. Conical rotation,
22. The laws of combination of the five symbols, t - - - 19
S, Y, K, T and U.
Remarks on notation. Biquaternions.
Examples to Chap. II. --------- 21
CHAPTER III.
FORMULAE AND INTERPRETATIONS DEPENDING ON
PRODUCTS OF VECTORS.
23. Vector as directed area. Moment of force. 23
24. Volume of parallelepiped = iSa^y. ------ 23
25. The formula, ....... 24
26. Resolution of a vector along three lines, ..... 24
pSa/3-y = aS/3yp + /3Sy ap + y Saftp,
pSa/?y = V^y Sap + VyaS/fy + Va/3Syp.
27. Resolution of a vector along and perpendicular to a given vector, 25
p = A . X- V = AS A- 1 /) -1- XVX-y
28. T(/3-a)2 = T/3 2 + 2Sa/3+Ta 2 . ... 25
Fundamental formulae of a plane triangle.
Examples to Chap. III. ........ 26
CHAPTER IV. '
APPLICATIONS TO PLANE AND SPHERICAL TRIGONOMETRY.*
29. Coplanar versors. De Moivre's theorem, ..... 27
SA
U^= cos A + L sin A = ** .
Quaternion as a power of a vector q a*.
Roots of a quaternion. Exponential, power and logarithm of a
quaternion.
*The beginner may pass at once to Chapter V.
xii CONTENTS.
ART. PA<Jfc
30. Spherical trigonometry. The fundamental trigonometrical rela- 29
tions for a spherical triangle deduced from the identity
<x y
31. The quaternion ftor l -y and its interpretation. The area of a 31
spherical triangle is twice the angle of the quaternion
Examples to Chapter IV., 33
CHAPTER Y.
GEOMETRY OF THE STRAIGHT LINE AND PLANE.*
32. Relations between a straight line and a plane. The method of 35
equations :
S(/3-y)a=0 for a plane, V(p-8)=0 for a line.
The method of indeterminates :
p=y-f ax for a plane, p=S+ftt for a line,
r being an arbitrary vector subject to an implied condition
and t being an arbitrary scalar.
33. The line through two points and a plane through two points. - 37
34. The plane through three points. Plane satisfying given con- 38
ditions.
35. Intersections of planes and conditions of intersection. 39
36. A pair of lines. Shortest vector perpendicular. Points of 39
closest approach.
Examples, complex, congruency, ruled surface.
37. Anharmonics. .. 41
38. Symmetrical relations for a tetrahedron. ----- 42
39. Relations connecting five vectors. 43
40. Hamilton's anharmonic coordinates. 43
Examples to Chapter Y. - - 45
*The last three Articles may be omitted at first reading.
CONTENTS. xiii
CHAPTER VI
THE SPHERE.*
ART - t PAGE
41. Equation of a sphere, intercepts on a line, tangent cone, poles 49
and polars.
42. Two spheres, radical plane, angle of intersection. 50
43. Three spheres, radical axis. Four spheres, radical centre. - - 51
44. Inversion. 52
45. Examples relating to a tetrahedron and a sphere. 53
46. Product of vector sides of inscribed polygon. 55
47. Inscription of polygons. 56
Examples to Chapter VI. 58
CHAPTER VII.
DIFFERENTIATION^
48. Differentiation of a vector which involves a single variable 62
parameter. o?=p=d>(t)
represents a curve. The derived vector
t'= t-t &=<) h
is a tangential vector. The differential is
dp =pQ.d= <'(*). dzf.
49. The equation. 64
op*p=<(*,)
represents a surface. Tangential vector, normal vector,
tangent plane, equation of normal.
.50. The equation p=< Of, M, v) and its interpretation. 65
.51. Differential of a function of a quaternion, 66
d. JPfo)-lim n \F(q+^)-F(q)}=f(W.
n=w >. V. I* f J
Distribution property of the function /.
.52. Differential of a function of several quaternions. - - - 6V
* Arts. 44-47 may be omitted at first reading.
f Articles 58-61 and possibly Arts. 55-56 may be omitted at first reading.
xiv CONTENTS.
ART. p AGK
53. Differentials of S0, V?, K?, Tg, U#. ..... 68
= .
T? q Uq q
54. Differential of a scalar function P of a vector p, - - - - 69
d/'=-Svdp, d'P=-Svd'p, d"P=-Svd"p,
wliere dp, d'p, and d"p are three non-coplanar differentials of
p. Introduction of Hamilton's operator V,
_ V<fyd> dP+Yd"pdp . d'P+Ydpd'p . d"P^ - p
v ~ Sdpd'pd'p '
55. The result of operating by V on a quaternion function of p is - 70
where dv is an outwardly directed element of vector area of
any small closed surface surrounding the extremity of p and
where v is the volume included by the surface.
Illustrations from fluid motion. Equation of continuity.
56. For any small closed circuit at right angles to an arbitrary 72
vector TJi', i r
Y (UvV) . Fp=\im ~ J dp^p,
where J. is the area enclosed by the circuit and dp a directed
element of length of the circuit.
Condition for perfect differential,
VVo-=Oif Scrdp=dP.
57. Analytical expressions for V ; one being ..... 74
The operator V may be treated as a symbolic vector ;
both when q is the operand and when it is a constant
quaternion.
58. Given p=F(q\ to find &q in terms of dp. Particular cases 77
solved, the general theory being explained in Art. 150.
Differential of 91 th root of quaternion.
59. Successive differentiation and development. Taylor's theorem 78
for quaternions,
f(q+P)=fy+df<2 + d % + etc., where dq=p.
CONTENTS. xv
ART. PAGE
60. Successive differentiations corresponding to different differentials 79*
dg, d'<?, etc., of q.
61. Stationary value of a scalar function of p with or without 80
equations of condition.
Examples to Chapter VII. ........ 82.
CHAPTER VIII.
LINEAR AND VECTOR FUNCTIONS.
62. Symbolic definition of linear vector function, .... 88
where a and ft are arbitrary vectors.
Trinomial form for the function and its conjugate <',
<p A'S Ap + //S/AP + i/S vp, <'p = AS A'p -t- juiVp + vSi/p.
63. Considered geometrically the equation ..... 90
<r=4>p
establishes a linear transformation from vectors p to vectors
o-, equal vectors becoming equal vectors.
64. To pass from vectors cr to vectors p there is the inverse trans- 90
formation p=z<j>~i<r or wp^^cr,
where the scalar m and the linear vector function i/r depend
only on direct operations of <'.
65. Cases of exception. The auxiliary function x> and the in- 91
variants m f and m",
xVa0
m" = Sa/3y
m'SojSy
The symbolic relations
m = </>X/r = >/r^,
66. Tlie symbolic cubic
</> 3 - m"< 2 + m'</> - w 0,
and the latent (scalar) cubic
<7 3 - mV 2 + mfg - wi = 0.
The axes y 1? y 2 and y 3 of <#> and the latent roots g^ g t and g z .
-
if p
Axes of a function and those of its conjugate determine sup-
plemental triangles on a unit sphere.
Special cases, indeterminate and coincident axes.
xvi CONTENTS.
ART. PAGE
67. The self -con jugate part $ and the spin-vector , - - - - 96
The axes of a self -conjugate function < are mutually rectangular
and the latent roots are real.
68. The relations .......... 97
69. Geometrical interpretation of the vanishing of invariants. - 98
70. The square root of a linear vector function, 99
71. Miscellaneous theorems relating to linear functions. - - - 100
The canonical reduction of a pair of functions.
Examples to Chapter VIII. ....... 102
CHAPTER IX.
QUADRIC SURFACES.
72. Equation of the general central quadric, ..... 106
73. Lengths of radii. Asymptotic cone. Tangent cone. Normals. 107
74. Pole of plana Reciprocal of quadric. ..... 109
75. Principal axes of section. ........ 110
76. Conjugate radii. - - - ....... 112
77. Cyclic planes. Hamilton's cyclic form, - - - .- - 113
-1.
78. Tangent right cylinders. Hamilton's focal form, - - - 115
79. Generators of a quadric, ........ 116
p = 'Y -- or l <j>a. +ya.
0. Non-central quadrics, ......... 117
$1. Quadric cones. Sphero-conics, Hamilton's proofs of the 118
associative principle.
$2. Confocal quadrics, ......... 121
Tangent ones to confocals. ...... . 123
CONTENTS. xvii
AaT. PAGE
84. The elliptic equation of confocals 124
P~ */{($ +&)( < t>+y)( < t>+z)}ty where 2 =0, Sc<^6=0, Se< 2 = -1.
Examples on surface of centres, umbilical generators, etc.
Examples to Chapter IX. 326
CHAPTER X.
GEOMETRY OF CURVES AND SURFACES,
(i) Metrical Properties of Curves.
85. The method of emanants. The vector of rotation (t) of the 131
emanant (77) at a point (p) of a curve,
t=
The vector curvature of a curve,
dUd P=Y ay
dp dpTdp
The vector torsion^
dUVdpd* P d3p
TJYdpd 2 pTdp r Vdpd 2 p
The vector twist of a curve,
to vector curvature + vector torsion.
86. The unit vectors, a tangent, /3 principal normal, y bmormal, 134
-J- = erf sas vector curvature ; = a x a = vector torsion ;
^ = a^a + ^y = vector twist,
where suffixes denote differentiation with respect to the arc.
Expansion of vector to point of curve in terms of arc.
87. The developables connected with a curve. General expressions 135
for their planes, lines and cuspidal edges.
(ii) Ruled Surfaces.
88. Euled surface regarded as generated by a moving emanant line. 137
The rate of translation of the emanent is pi, where p is the
pitch or parameter of distribution,
Udp qdpTJiy
= b = b -T .
* L
Vector equation of line of striction,
xviii CONTENTS.
ART. PAGE
89. Normal and tangent plane for ruled surface. The involution for 139
perpendicular tangent planes,
2 , (UQC+UQC'=O).
(iii) Curvature of Surfaces.
90. Curvature of projection of curve. Vector curvature of curve 141
traced on surface resolved into component curvatures in and
at right angles to tangent plane ;
dUdp^ Sdvdp t 1 s vdUdp
dp dpvTdp v dp
curvature of normal section + geodesic curvature.
91. Surface represented by /(p)=cowsj?., ...... 142
<=<' if n is constant. In general Sv=0, where e is spin-
vector of <.
92. Equation for principal curvatures, ...... 142
Sv(<^ -CTi/)- 1 v=0.
Tangents to lines of curvature,
Curvature of normal section through dp,
(7= Ci 008*2+08 ain a Z if Udp=r 1 cosZ+T 2 sin^.
Surfaces generated by normals.
93. Second method for curvature. Measure of curvature, - - 144
VdUvd'Uv
6l62= Vdpd'p '
Gauss's theorem of the linear element.
94. Kinematical method. Moving system of tangents and normal. 145
Examples on geodesies, etc.
(iv) Families of Curves and Surfaces.
95. Family of curves, ......... 148
p=7?(*; ,&,,...).
Curves touching given surface or intersecting given curve.
96. Differential equation of surfaces met in n consecutive points by 148
curves of the family.
97. Equation of family of surfaces, ...... - 149
/(p; a, 6, c,...)=0.
Genesis of partial differential equations.
98. Analogue of Charpit's equations. ....... 151
CONTENTS. xix
CHAPTER XL
STATICS.
ART. PAGE
99. The vectors a being drawn to points of application of the forces /?, 156
3=ju= resultant moment at o, 2/3=A=resultant force ;
p is pitch of resultant wrench, GT the vector perpendicular on
central axis, y the vector to Hamilton's ce?itre, 2a/5 is total
quaternion moment.
100. Reduction of system of forces to two forces. - - - 158
101. Astatics. The linear function ....... 159
For astatic equilibrium
<=0, A=0.
Arrangements of central axes relative to the forces and relative
to the system of points of application.
102. Composition of wrenches. Equation of three-system of screws. 163
The cylindroid. Resolution of wrench into components on
six given screws.
103. Example on equilibrium of heavy chain] lying on a surface 166
and acted on by any forces.
CHAPTER XII.
FINITE DISPLACEMENTS.
104. Rotation followed by displacement. 168
Composition of successive displacements. Small displacements.
Screw of the displacement.
105. .Twist- velocity of body. Instantaneous screw. The general 169
equation of relative velocity. Fixed and moving axes.
Acceleration.
106 Rotation depending on two parameters. 173
107. Examples on the applications of the relation. - - - - 173
p
52
xx . CONTENTS.
CHAPTER XIIL
STRAIN.
ART. PAGE
108. Homogeneous strain. Vectors p changed to o-, where - - 177
o-=</>, ra>0.
Strain ellipsoid T<jr l <r*=r.
109. Shear, dilatation, rotation. Reduction of general strain to - 178
110. Lines altered in given ratio are parallel to edges of the cone, - 179
T<U/) = const.
Condition that inclination of lines should remain unchanged.
Effect of superposed rotation on axes of <.
111. Displacement along and at right angles to p , - - - - 180
Elongation quadric.
112. Non -homogeneous strain, - -181
QTsrs 6'(p/ QCT == Obdp. fpOL' == ~~ oCt v . O", ft := vOCtO".
Condition for pure strain,
Wo-=0, o-=VP.
113. Case of small strains, 182
CHAPTER XIV.
DYNAMICS OF A PARTICLE.
114. Equation of motion, 184
Rate of change of moment of momentum round fixed point or
moving point. Energy equation.
115. Fixed centres attracting according to law of distance. - - 185
Solution of equation of damped vibrations
and of the more general equation
P+<!P-I-< 2 P=0.
116. Central forces, .......... 186
Law of nature. Circular hodograph. Examples on moving
orbits, etc.
117. Constrained motion. ......... 189
118. Brachistochrone. ......... 192
CONTENTS. xxi
CHAPTER XV.
DYNAMICS.
ART. PAGE
119. For any number of particles, the reactions cancel in the equations 194
where M is total mass, p vector to centre of mass; 6 moment
of momentum at origin, resultant force, 77 resultant moment
at origin. Energy equation.
120. Moment of momentum with respect to moving point and with 195
respect to centre of mass.
121. Case of rigid body. Equations of motion, .... 19$
Mp=, <d> + V<o<a>=77.
Energy equation.
122. Instantaneous twist- velocity produced by impulsive wrench 200
acting on free body. Energy equation.
123. Case of constrained body. Reciprocal screws. Evoked and 204
reduced wrenches.
CHAPTER XVI.
THE OPERATOR V.
(i) The Associated Linear Functions.
124. The invariants and auxiliary functions for - - - 211
<a=-SaV.cr, <'a=-VSacr,
in which cr is a vector function of p.
When cr denotes the velocity of the extremity of p, the rates
of change of a line-element (dp), a surface-element (di/) and
a volume-element dv are
D t . dp = <dp, D t . di/ = x'dv, D t . dv = m"dv.
The quaternion invariant
m"-%e= -Vcr ;
m" = - S Vcr = divergence, 2e = V Vcr = curl.
The auxiliary function ^
T/ry - W V'Scrcr'y , Vy = - Sy V V . Vcrcr 7 ;
and the invariants
xxii CONTENTS.
(ii) Integration Theorems.
ART. PAGE
125. The transformations ........ 215
Cases (A) of discontinuity ; (B) when q is multiple-valued ; (c)
when q becomes infinite ; (D) of multiply-connected region.
(iii) Inverse Operations.
126. Interpretations for the functions ...... 218
P=zV-iq and r=V- 2 2 where Vpq and W 0,
deduced from the identity
vy.d.-
' V
(iv) Spherical Harmonics.
127. Expansion in terms of spherical harmonics. - 222
The fundamental theorems.
(v) Various expressions for V.
128. Expressions for V and V 2 in terms of arbitrary differentials of p. 225
Case in which p is given as a function of three parameters.
Examples on systems of equipotential surfaces, etc.
(vi) Kinematics of a deformdble system.
129. Bate of change of quantity q associated with point moving 228
with velocity cr,
The relations
D t (qd.v)=(D t q+m"q).d.v, D(Strrdv) = S^dv, D t S&dp = Sjgdp,
where W==GT~VVVor57-o-SV57, g = CT~ VScrST -
The voluminal, areal and linear equations of continuity
Euler's and Lagrange's methods.
130. Flow of a vector GT along a curve, and rate of change of flow, - 231
Circulation of the vector for a closed curve,
CONTENTS. xxiii
ART. PAGE
Flux of a vector T through a surface
131. Expression for vector 57 in the form ..... 233
G^VP+VTj+V.S where V 2 ^=0.
Irrotational distribution if 77=0 ; no divergence if jP=0.
Transformation relating to vortex motion.
132. Transformations effected by means of the invariant property 235
of V.
J or . dv = J pSVsr . dv - j pSdvo? = WpYVtrr . dv - W" pVdyor,
and other analogous relations.
(vii) Equations of motion of a deformable system.
133. Equations of motion for Euler's method, ..... 236
and for T.;: i: ; . , _> ".- method. Equations of continuity.
134. Determination of stress function <3? for viscous fluid and for 238
isotropic solid.
135. Rate of change of kinetic energy. ...... 239
136. Dissipation function for viscous fluid. ..... 240
Example on motion of solid in liquid.
137. General case of elastic solid. Equation of motion, ... 242
Quaternion statement of Hooke's law,
3>a=9(a, V, (9)=9(a, 6, V).
Energy function. Elastic constants.
138. Equation of vibrations of elastic solid, ..... 247
o^0(V, V, 8).
Equation for plane wave moving with wave-velocity i>,
rf-eg. ^ *).
Three plane polarized waves propagated in direction Uv with
vibrations parallel to axes of function 6(Uv, Uv, a). Wave-
velocity surface. Internal conical refraction. Wave-surface
as envelope of
S=l or
Eay-velocity />, c/>= -6 u0, U0, .
xxiv CONTENTS.
(viii) Electromagnetic Theory.
ART. *AGK
139. The circuital laws, ......... 249
where 77 is magnetic and e electric force ; y electric and y,
magnetic current, and ^ velocity of light. Differential
equations of field,
where 5 is electric displacement, /3 magnetic induction ; L
electric, i,. magnetic, conduction current; e density of elec-
trification carried with velocity v and e, density of magneti-
fication carried with velocity v,.
Meaning of and 17 ; e* and f] t are total forces ; e* and 77* are
impressed forces ; and
Conduction current equations
* = *( + ), t,
Displacement and induction equations,
140. Activity of impressed electric and magnetic forces. Evoked 251
mechanical force and stress on element of medium.
Joulian waste of energy due to resistance
/=-S^-i6-St / <i> / - 1 V
Stored energy, electric ( W) and magnetic ( W f ) ;
Radiation of energy. The Poynting vector uVerj.
Determination of evoked mechanical force () and of stress
function <iv Across an arbitrary vector-area /A, the stress is
141. Explicit equation for e when there is no convection current and 255
when circuit is at rest. Propagation of disturbance in di-
electric and in conductor. Case of no applied forces. Normal
solutions.
142. Propagation of light in crystalline medium on Clerk Maxwell's 256
hypothesis. The equations
where v is wave-velocity. The implied relations
CONTENTS. xxv
PAGE
The ray-velocity (p) and the five vectors 8, /?, u, e and rj are
connected by the relations
Determination of the vectors when one is given. Pair of plane
polarized waves with given direction of wave- or of ray-
velocity. Relations connecting the vectors depending on the
two waves. Construction for the vectors by means of two
quadric surfaces.
Conical refraction. Wave- and ray-velocity surfaces.
CHAPTER XVII.
PROJECTIVE GEOMETRY.
143. A quaternion (q) represents a point (Q) loaded with a weight Sg ; 263
The sum of weighted points is their centre of mass loaded with
the sum of their weights.
144. The combinatorial functions ....... 264
(a, 6) = 6Sa-aS6 ; [a, 6]=V . VaVfc ;
[a, 6, c] = (a, ft, c)-[ft, c]Sa-[c, a]S6-[a, b]Se ; (a, 5, C ) = S[a, ft, c] ;
(a, b, c, d)=aa[b, c, <%
Symbol of plane [a, 6, c] ; principle of reciprocity.
145. The equations .......... 266
CSS a ? 6]=0 and (#, a, 6, c)=0
represent the line ab and the plane abc.
The plane S^^=0 and its reciprocal with respect to the unit
sphere S. 2 =0.
Formulae of reciprocation,
([060]; [abd])=[ab](abcd)' t [[abc] ; [abd]]= ~(ab)(abcd).
146. The relations connecting five points, ..... 268
a (bcde) + b (cdea) + c (deal) + d(eabc) + e (abed) = ;
e (abed) = [bed] Sae - [acd] Sbe + [abd] See - [dbc\ Sde.
147. Combinatorial functions. Construction and development of 270
these functions.
ART.
CONTENTS.
PAGE
148. The general linear transformation in space, - 272
P=fy-
Determination of transformation converting five given points
into five others.
149. When /transforms points,/- 1 transforms planes. 273
150. Inversion of a linear function. The auxiliary functions, 273
F, Cf, H ; the invariants TZ, n', ?i", ri".
151. The united points, lines and planes. ...... 274
152. The self -con jugate and the non-conjugate parts of a function, - 275
The equations of the general quadric and of the general linear
complex are respectively
The equations of the polar plane of a and of the plane contain-
ing the lines of the complex through b are respectively
and the equations of the reciprocals of the quadric and of
the complex are
83/0-^=0, Sp/r^-o.
Nature of united points of/.
Common self -con jugate tetrahedron of two quadrics
is determined by united points of fi" 1 /^
Examples on generalized confocals, etc.
153. Square root of linear quaternion function. - 282
.Reduction of function to form
/=/,/, where /,=(//)*, /./.'-I.
Further reduction of /,
Transformations converting one quadric into another, etc.
Curve of intersection of two quadrics.
Examples on lines traced on surfaces, etc.
154. Invariants of linear transformations and of quadric surfaces. - 288
155. Numerical characteristics of curves represented by - - - 290
l>i^ 3 !M and by
number of points represented by
(prf^Q and by
where p n is homogeneous and of order m n in q.
CONTENTS. xxvii
ART. PAGE
156. The general surface. The relations of reciprocity, - 293
Conjugate tangents, asymptotic lines, generalized curvature.
157. Poles and polars. Operator D analogous to V. Avonhold's 296
notation.
Examples on Jacobians ; surface through points on given
surface where tangent touches in four consecutive points.
Examples to Chapter XVII. ....... 300
CHAPTER XVIII.
HYPERSPACE.
1-58. Extension of formulae to any number of variables. - - - 303
Definition of product of two vectors in w-space ;
where V 2 a/3 is 'vector-area, of parallelogram, and V a/3 is Sa/3.
Multiplication in general denned to be associative and dis-
tributive. Product of m vectors,
Expansion of
159. Sum of area vectors not generally an area vector but the 306
analogue of an angular velocity. Kotation in w-space.
160. Symbols of points and flats in hyperspace. Formulae for 308
protective geometry.
INDEX, ........... 310
CHAPTER I.
THE ADDITION AND SUBTRACTION OF VECTORS*
ART. 1. A right line, AB, considered as having not only
length but also direction, is said to be a vector. The direction
of the vector AB is that of the point B as viewed from A, and
the vector BA is the opposite of AB, being equal to it in length
but having the opposite direction. All equal right lines AB,
A'B', etc., which have the same direction are equal vectors.-}*
ART. 2. The sum obtained by adding the vector BC to AB is
denoted by BC+AB, and is defined to be the vector AC. Thus
symbolically (fig. 1),
BC+AB = AC.
D C
A ' ' 3 A B
FIG. 1. FIG. 2.
Completing the parallelogram, ABCD, the definition of addition
gives likewise the equation (fig. 2)
DC+AD = AC
or AB + BC = AC,
because the vectors DC and AD are respectively equal to AB and
BC. Thus the sum of two vectors is independent of the order
, * Following the example of Hamilton in his Lectures on Quaternions and in his
Elements of Quaternions, the tahle of contents of this volume is amplified into an
analysis or commentary to which it may be useful occasionally to refer.
t It seems to be an unnecessary complication to print a bar ( AB) over the letters
which represent a vector AB, Hamilton sometimes uses the notation AB to re-
present the length of the vector AB.
J.Q. A
2 ADDITION OF VECTORS. [CHAP. i.
in which they are added, or the addition of two vectors is a
commutative operation.*
ART. 3. The sum obtained by adding any vector CD f to the
sum of AB and BC (fig. 3), is the sum of CD and AC, or the
vector AD. But AD is likewise the sum of AB and BD, that is,
the sum of AB and the sum of BC and CD. And by completing
--.C
FIG. 3,
the parallelogram of which BD is a diagonal and BC and CD are
sides, it appears that AD is also the sum of BC and the sum of
AB and CD. In other words, the same vector is obtained by
adding any one of the three vectors, AB, BC and CD, to the sum
of the other two. This vector sum AD is consequently inde-
pendent of the order in which the component vectors are taken
and of the mode in which they are grouped.
The same process applies in general, and the addition of
vectors is an associative and a commutative operation! It is
associative inasmuch as the vectors may be grouped into partial
sums in any way ; and it is commutative because the order in
which the vectors are taken is immaterial.
ART. 4. Any number of vectors being arranged as the succes-
sive sides AB, BC, etc., of a polygon, their sum is the vector AD
drawn from the initial point of the first to the terminal point of
the last. If the polygon happens to be closed, the sum is a
vector of zero length, or simply zero. Thus, in particular,
AB+BA = 0, AB+BC+CA = 0, AB + BC+CD+DA = 0.
ART. 5. It is natural, in accordance with the equations just
given, to introduce the sign , and to write
*In certain systems of vector analysis, the word vector is used in a different
sense, and a vector cannot be determined without reference to its position. The
commutative law then ceases to be obeyed. An example of non-commutative
addition will be found in Art. 21, p. 16.
t In every case, unless the contrary is expressed or implied, the vectors with
which we deal are not necessarily parallel to a plane.
ART. 6.] SCALAB COEFFICIENTS. 3
or to agree that the sign prefixed to a vector shall convert it
into its opposite (Art. 1). Hence the subtraction of one vector
from another may be regarded as equivalent to the addition of
the opposite of the first vector to the second. Subtraction of
vectors is thus included in addition.
As we can now interpret AB, it is convenient to use a single
symbol to denote a vector. We shall follow Hamilton's admir-
able notation, and shall employ the small letters of the Greek
alphabet to represent vectors, using, as a general rule, the earlier
letters a, /3, y, etc., for given or constant vectors, and p or <r for
variable vectors.
ART. 6. The sum of two equal vectors is a vector of the same
direction and of twice the length. It is natural to write, as in
algebra, 2 a = a+ct, 3a = a+a+a, etc.,
and generally, at least when n is an integer,
if the vectors /3 and a have the same direction while the length
of /3 is n times that of a. This result may be extended to the
case in which n is fractional or incommensurable by a process
identical with similar extensions in elementary algebra. The
last article affords the interpretation to be adopted when n is
negative; and when n is complex (ri+\/ ITI"), the difficulties
of interpretation are of the same nature as in ordinary algebra,
and need not be discussed here.
Further, it is natural to say that the coefficient n results from
the division of the vector /3 by the parallel vector a, and we
shall therefore write
Q
7i = S or 7i=/3-7-a, or 7& = /3:a,
a
as a consequence of fi~na. Also, conversely, whenever the
quotient of two vectors is an algebraic quantity or a scalar?
we infer that the vectors are parallel, and that they have the
same or opposite directions according as that scalar is positive or
negative.
Again, if n is an integer and if a and /3 are any two vectors,
the laws of addition give
and by a process of induction this relation may be extended to
*The word 'scalar,' synonymous with algebraic quantity, was employed by Ij
Hamilton because such a quantity may be conceived to be constructed by " com- j
parison of positions upon one common scale (or axis)." Elements, Art. 17. \\
4 ADDITION OF VECTORS. [CHAP. i.
the case in which -n is fractional or incommensurable. More
generally, if a?, y and z are any scalars,
so that the multiplication of vectors by scalars is a distributive
operation.
ABT. 7. In the calculus of quaternions a unit of length is
selected to which the lengths of all vectors are referred. The
tensor of a vector a is the number of units contained in its
length, and is denoted by- the symbol Ta. Thus the tensor is a
positive or " signless " number, at least when the vector is real,*
&nd in particular, m = m / _ \
In general, if n is a real scalar,
Tna^nTa if ^>0; Tna=-nTa if ^<0.
Hamilton also uses the notation Ua to denote a vector of unit
length having the same direction as a, and he calls Ua the versor
of the vector a. Since the direction of a is opposite to that of a,
Ua=-U(~a),
and, more generally,
Ufta=Ua if >0; U^a= Ua if ??,<(),
Also, by Art. 6, a = Ta . Ua,
or a vector is the product of its tensor and its versor.
FIG. 4.
ART. 8. An arbitrary vector OD (or <5) may be resolved in one
way into a sum of vectors parallel to th/ee given and non-
coplanar vectors OA, OB and 00 (or a, ft and y).
*For imaginary vectors see Art. 22, p. 20.
ART, 8.] EXAMPLES. 5
Through D draw three planes parallel to the planes BOO, COA
and AOB, meeting the lines OA, OB and 00 in the points A 7 , B', C'.
Then it is evident from the figure that
OD==OA'+OB'+OC'; or
or =
if the scalars x, y and z are the quotients of parallel vectors,
o? = OA':OA, 2/ = OB':OB, 2=OC':OC;
and it is further evident that this construction is unique.
It may happen that some or all of these three scalars are
negative, or some may be zero, but these cases can present no
difficulty.
Ex. 1. Find the vector oc to a point which divides AB in a given ratio.
r AC CB AC+CB ft-q y-a fa+m{i~]
L Here m^T-TnT"!?^ m> or ?"7+5rJ
* Ex. 2. If weights I, m and n are placed at A, B and c, find their
centre of mass.
~ [The extremity of the vector (Zq+mj3+y) :(l+m+n), supposed to be
coinitial with a, p and y.]
* Ex. 3. Prove that the mean centre of a tetrahedron is (a) the intersection
of bisectors of opposite edges ; (6) the intersection of lines joining the
vertices to the mean points of the opposite faces. Show that the former
lines bisect one another, and that the latter quadrisect one another.
* Ex. 4. Prove that the vectors a/3j when drawn through a common
point terminate at the vertices of a parallelepiped.
Ex. 5. Discuss the arrangement of the extremities of the sixteen coinitial
vectors aByS. Consider the points with reference to the extremities
of q, etc., and with reference to one of the points, the extremity ot
a _|_ ^g 4. y + f or example.
Ex. 6. Prove that four arbitrary vectors are connected by a linear
relation, aa,+bp+cy+dS=Q.
Ex. 7. If three vectors are linearly connected, or if
they are coplanar.
Ex. 8. If aoA+6oB+coc=0, a4-6+c=0, the points A, B, c are collinear.
Ex. 9. If aoA+&oB+coc+tfoD=0, a+b+c+d=Q, the points A, B, c, i>
are coplanar.
CHAPTER II.
MULTIPLICATION AND DIVISION OF VECTOES AND
OP QUATEEJSTIONS.
ART. 9. The product of the length of one vector (a) into the
length of the projection of another (/3) upon it is denoted by the
expression rat
x iaap, w uotp = o/frc^cp
and this function Sa/3 of two vectors is called the scalar of a/8.
By similar triangles it follows that (fig. 5)
FIG. 5.
and because the sum of the projections of any number of vectors
on any line is the projection of their sum, it appears that (fig. 6)
and therefore the function is a doubly distributive function, or
If the vectors a and y are at right angles,
Sqy = >
and conversely.
An equation such as Sa/8 =
implies that the projection of a on /3 multiplied by the length of
/3 is equal to the projection of y on 8 into the length of S.
ART. io.] SCALAR AND VECTOR. 7
ART. 10. A unit of length having been assumed, let a vector f
be drnxvrr at rjjrht angle* to (wo jri\\- r i vc-otors a andjS so that
rotation imuid ihi< \v<-:or fivin "io ,3 i- positive,* anct let the
length of this vector be numerically equal to the area of the
parallelogram determined by a and /3. This vector is denoted
by the symbol Va/3,
and is called the vector of a/3.
If the vectors are taken in the reverse order, VjSa has the
same length as Va/3, but the direction is opposite, the rotation
being now reversed, so that
V/3a=-Va/3.
If an equation such as Va/3 =
FIG. 7.
exists, the vectors a, /3, y and 8 must all be parallel to the same
plane; the areas of the parallelograms determined by a and ft
and by y and S must be equal, and the sense of rotation from
a to /3 must be the same as that from y to S (fig. 7).
Like Sa/3, the function Va/3 is a doubly distributive function.
If /3' is the component of the vector /3 at right angles to a it is
obvious that
and the tensor of Va/3 is equal to the product of the tensors of a
and of & (fig. 8).
* The convention respecting rotation which is here adopted is the opposite of
that employed by Hamilton. The axis of a rotation is taken to be in the direction
of the advance of a right-handed screw turning in a fixed nut, and this system is
now known as the right-handed system of rotation (Clerk Maxwell, Electricity and
Magnetism, Art. 23). On the other hand Hamilton calls his system right-handed,
but he takes as the axis the direction from blade to handle of a turn screw when
screwing a right-handed screw into a nut (Lectures, Art. 68, Elements, note to
Art. 295), and accordingly some little care is necessary in comparing Hamilton's
demonstrations with those of the present volume. Tait uses the modern right-
handed system in his quaternion writings.
MULTIPLICATION OF QUATERNIONS. [CHAP. n.
If /3' and y' are the components of ft and y at right angles to
and in the plane of the paper while a is drawn upwards at
a, ana in tne piane 01 uae paper wane a i m-ttwu. ujjweiru
right angles to the plane (fig. 9), the vectors Va/3' and Vay' will
TB'
FIG. 9.
lie in the plane of the paper, at right angles respectively to ft'
and y'. But TVa/3' : T/3' = TVay' : Ty' = Ta, and consequently the
triangles OB'C' and OB^^ are directly similar. Hence OC X is at
right angles to 00' and 'TOO, : TOG' = Ta. Consequently
In this relation we may replace ft' and y' by /3 and y, so that
a, /3, and y being three arbitrary vectors.
We have now
V2a2/3 = SSVa/3
for any number of vectors, since in particular for four vectors,
If yqj6=0 without having either a or /3 zero, the vector a
must be parallel to /3, for the area of the parallelogram deter-
mined by a and. f3 must vanish.
ART. 11. The product of the vector a into
6( l nation > q^Saft + Vaft
is defined by the
(B)
and because it is the sum of two doubly distributive parts, it is
likewise doubly distributive, or '
The product pa is not generally equal to aft. In fact
' /3a = Sa / 8-Va/3 because Sa/3'=S/3a, V a /3= -Y/3a,
ART. is.] PRODUCT OF TWO VECTORS. 9
Thus multiplication of vectors is not commutative. We speak
of a/3 as the product of ft by a, or the product of a into /3.
Adding and subtracting the expressions for the two products
a/3 and /3a, we find
, Va/3 =
ART. 12. The sum. of a scalar and a vector is called a quater-
nion because it involves four independent numbers, such as the
scalar and the three coefficients of the vector when resolved
along three given directions (Art. 8).
Thus the product of a pair of vectors is a quaternion, and
conversely, every quaternion may be expressed as a product of a
pair of vectors. If q is a quaternion, if Sq is its scalar part and
Vq its vector part, so that
Jf a and /3' are two vectors at right angles to one another and to
V g , so that Va/3' = V g ; and if /3 /3' is the vector parallel to a,
for which Sa(/3 /3') = S g , then we have
V g = Va/3 because Va(/3-/3') = ; S g = Sa/3 because Sa/3^0,
and therefore q = a/3,
or the quaternion has been reduced to the product of a pair of
vectors.
Scalars and vectors may be regarded as simply degraded cases
of quaternions.
The sum of any number of quaternions we define to be the
sum of their scalar parts plus the sum of their vector parts.
Addition of scalars is associative and commutative, and likewise
addition of vectors (Art. 3). It follows that addition "of
quaternions is associative and commutative.
/ ART. 13. We next define the product of a quaternion and a
vector to be distributive with respect to the scalar and the vector
of the quaternion. Thus
The products yVq and V g . y fall under formula (B), and we
define that multiplication of a scalar and a vector is commutative,
so that yS g = S g . y.
Thus we can interpret expressions such as a . /3y or a/3 . y (the
product of a into the product /3y and the prpduct of, the pro-
duct a/3 into y), and we see that they are distributive with
respect to the three vectors, so that
. /3y,
10 MULTIPLICATION OF QUATERNIONS. [CHAP, n.
We shall now prove that the products are associative^ so that
we may omit the points, and to this end we shall consider the
laws of combination of three mutually rectangular unit-vectors,
i, j and k.
ABT. 14. Let any three mutually rectangular unit-vectors,
i, j and k, be drawn so that rotation round i from j to k is
positive.
According to the usual convention, if i and j are in the plane of
the paper, k will be directed vertically upwards, and it is seen at
J'*
Fio. 10.
once that rotation round j from k to i, and also round k from i
to j is positive (Fig. 10).
We have then, because the vectors are mutually perpendicular
and of unit length,
i 2 =S? 2 = S& 2 =-l; .......... (Art. 9)
c? = -i, Vi&= -j, V?i= -7c; (Art. 10)
and by formula (B) it follows at once that
'i 2 =j 2 =7c 2 = 1, jk*= i=kj, ki=j=ik, ij=sk**ji... . (c)
Let us now, as in the last article, form the ternary products of
these vectors. We have by the relations just given
i ,jlc=i . i=* 1 =7t3 . kij . k*=*ijk t
the points being omitted as they are seen to be unnecesHary.
Similarly, for every ternary product of i, j and k, the points may
be shown to be unnecessary.
For quaternary products, let *, K , X, JUL each denote some one of
the three symbols i, j, k, then
I . K . X/X = UC . \JUL-IK . X . JUL = tK\ . /A
because, for example, L . K . X^t is a ternary product, as X^u must be
i> y, k or -1. In this way all products of the symbols
i, j, k are seen to be associative.
ART. 16.] ASSOCIATIVE PROPERTY. 11
It may be a useful exercise to show that the associative law
enables us to deduce all the relations (c) from Hamilton's funda-
mental formula (A), . 2 . 2 72
v ^ 2 =/ = A; 2 =^jfc=--l ......................... (A)
For example, i . i;%= i gives jk=i.-
* Ex. 1. Prove that
Ex. 2. If the symbols i, j, k obey the laws,*
p_j.k"=+l; jk-i, ki=j, =k; W--1, ik=-j, ji=-k,
prove that their multiplication is dissociative.
[i 2 .j = +j but i.ij=i.k=-j.]
ART. 15. We can now show that multiplication of vectors is
associative. Let any three vectors, a } ~$ and y be expressed in
terms of i,j 3 &, so that
By Art. 13,
a . /3y = SSSari .
a/3 . y = 222 ajij/y . ^ft =
so that a . /3y = a/3 . y = a)8y,
and similarly for all products of higher orders.
Hence multiplication of quaternions is associative, for a qua-
ternion may be expressed as the product of a pair of vectors;
It now appears (compare Art. 13) that the product of any
number of vectors taken in any given order is a definite
quaternion.
ABT. 16. The division of vectors may be reduced to multi-
plication. By formula (B) the square of a vector is
a^S.q^-ra 2 ; so that a<-2 = l >
and thus it appears that - a : (Ta) 2 is the reciprocal of the vector
a, say cT 1 or ~ The vector cr 1 is opposite to a in direction,
* Mr Oliver Heaviside bases his rectorial Algebra on these laws. Prof. Knott
(Kecent Innovation in Vector Theory, Proc. R.^.E., 1892-3) draws attention to
papers written by the Rev. M. O'Brien in the years 1846-52, in which the square
of a -vector is taken to be positive.
12 MULTIPLICATION OF QUATERNIONS. [CHAP. n.
and its tensor is the reciprocal of that of a. We can therefore
interpret products such as
/3a~ l and a~ l /3,
and the first of these we shall call the quotient of by a, and
denote it by o
or
a
The reciprocal of any product of vectors is the product of their
reciprocals taken in the reverse order. Jb'or if
we have QQ' = 1
in virtue of the associative law. Similarly, the reciprocal of a
product of quaternions is the product of the quaternions taken
in the reverse order! Hence every quotient of vectors or of.|
quaternions is a quaternion ; and more generally every com- j
bination of quaternions by the processes of addition, subtraction*
multiplication and division is a quaternion. *
Ex. 1. Prove that
y Ex. 2. Distinguish between the expressions
S 'ft : , 8/5
y a w ya'
[These may be written By^ftar 1 and S/Ja^y 1 .
/ Ex. 3. Prove that
ay a ' ya
1 ^ART. 17. The conjugate Key of a quaternion q is defined by
I the relation ^ = S^-V/7.
I-
If then q = a/3, we have K# = /3a (Art. 11), and
qKq = a/Spa = a 2 /3 2 = Kqq = Ta 2 T/3 2 (Art. 16).
The products of the tensors of the vectors into which a quaternion
is resolvable is therefore independent of any particular selection
of the vectors since Sg and Vg are independent of any particular
pair of vectors ; and the ^square of this product is
if we call this constant product of tensors, the tensor (T<?) of
the quaternion.
ART. is.]
Again,
TENSOB AND VERSOE.
13
and TJq = UaU/3 is called the versor of the quaternion. If TJT L.q
is the angle between the vectors a and /3, which is less than two
right angles and measured from a to /3, we see by the definitions
of Sq and Vg that (Arts. 9 and 10)
The angle Lq is called the angle of the quaternion, and is
independent of any particular set of vectors a, /3.
A plane at right angles to Vg is called the plane of the
quaternion and UVg is called the axis.
Ex. 1. Prove that - Kg' ^w-ix ~jy - kz t
where q
Ex. 2. Write down the analogous functions of Kq in terms of #, y,
and -z^.
Ex. 3. Prove that a^ 1 /? = K . /3cr x .
Ex. 4. What is the nature of q if
If #= - Kq ?
ABT. 18. We can always reduce a quaternion to a quotient of
vectors (Arts. 12, 16), and write
B OB m TOB TT UOB OA 7 T7 A'B
the line'BA' being drawn perpendicular to OA.
Thus the^shage - v of the triangle AOB is constant for a given
quaternion. * From this point of view, a quaternion is called by
Hamilton a ratio of vectors, as it depends on their relative
magnitudes and on their relative directions.
14
MULTIPLICATION OF QTJATEBNIONS. [CHAP. n.
It is not difficult to show that the conjugate (see Fig. 12)
Tr OB' .. OB
for
The triangle AOB' is inversely similar* to AOB.
FIG. 12.
ART. 19. Conversely, if the product qa is a vector /3, it is
evident that a and /3 are both at right angles to Y#. And if a
is any vector at right angles to Vg, qa is a vector making a
constant angle (LC[) with a, and having its length Tq times that
of a. In other words, regarding the quaternion as an operator,
it turns vectors in its viaine through a given angle, and alters
their lengths in a given ratio. In particular we may regard a
vector as turning vectors at right angles to it through a right
angle, and altering their lengths proportionately to its own.
The versor U# turns vectors in its plane through the angle Lq
but leaves their lengths unaltered. The tensor Tq alters the
lengths of all vectors in a given ratio. The total effect produced
by q on a vector in its plane may be considered to be effected in
two stages or at once as indicated by the relation
ART. 20. The results of articles 18, 15 and 16 afford an ex-
. tremely elegant construction for the product of two quaternions
q and r. Take any vector OB along the line of intersection of
the planes of the two quaternions. Make the triangle BOG in
* Hamilton uses the phrases direct similitude and inverse similitude in the sense
that two directly similar figures in a plane appear to have the same shape ; while
of two inversely similar figures one has the same shape as the reflection of the
other in a mirror.;
AET. 20.]
CONSTKUCTION FOE PEODUCT.
15
the plane o r similar to the triangle determined by r (Art. 18) ;
make AOB in the plane of q similar to the triangle of q ; then,
by the associative principle (Fig. 13)
- -<-
FIG. 13.
PIG, 14.
If the triangles BOA' and C'OB are respectively coplanar with
and similar to AOB and BOG, the second product is (Fig. 14)
, = OA.Y_OA' OB\
Ex. 1. Prove that JL(rq)='KqKr,
[Take c, on oc and A, on OA so that C,OB and BOA, are inversely similar to
BOO and AOB, and the triangle A / OO / is inversely similar to COA. Art. 18.] ;
" Ex. 2. The product of the conjugates of any number of quaternions is
the conjugate of their product in reverse order.
" [By Ex. ^
, 3. Show that
Ex. 4. If a 1 tt 2 ...a n are n vectors, and if ILa =a 1 tt 2 ...a 7l , TL'a=<i n a n -i.
show that
Ex. 5. Prove that
Spg-Sgy; T^pq^TVgpi L.pq=Lgp.
' Ex. 6. Prove that pKq+qKp=ZS.p'Kq = 2S. qKp.
1 Ex. 7. Prove that the tensor of a product of any number of quaternions
is independent of their order.
16 MULTIPLICATION OF QUATERNIONS. [CHAP. n.
* Ex. 8. Prove that the versor of a product of any number of quaternions
is the product of the versors taken in the same order.
Ex. 9. Show that three quaternions cannot in general be reduced
simultaneously to the forms
Ex. 10. Prove that the scalar of a product of any number of quaternions
is unchanged when the quaternions are cyclically transposed.
Ex. 11. Prove that the tensor of the vector part of a product of
quaternions remains unchanged for cyclical transposition.
Ex. 12. Prove the identity
(ww 1 xx' yy' 22)* + (ma?' 4- w'x -\-yz' y'zf
f (wy' + w'y 4- zsf - dxf + (wtf + itfz +xy f
[See Ex. 1 of this series and Ex. 1, Art. 1 7. This identity is of historical
interest as regards the discovery of quaternions. See Gravea's Life of Sir
William Rowan Hamilton, vol. ii., p. 437.]
ABT. 21. The multiplication of versors, to which the multipli-
cation of quaternions may be reduced by separating the tensors,
admits of a simple spherical representation.
FIG. 15.
A versor is represented by a directed great circle arc belonging
to a definite great circle (the plane of the versor) and having a
definite length (the angle of the' versor). From the figure
{Fig. 15)
OC 00 OB
OA' OA' OB
The spherical triangles ABC and A'BC' are inversely equal.
The construction recalls the construction for the sum of vectors,
it is allowable to write
ART. 21.]
SPHERICAL REPRESENTATION.
17
This addition of vector-arcs is not commutative, for CfAf is not
generally equal to AC equality of these vector-arcs requiring
equality of length, similarity of direction and coplanarity.
Two quaternions are commutative in order of multiplication j
if, and only if, they are coplanar. A necessary condition for *
commutation is that the arcs AC and A'C' should belong to the
same great circle. If OB is not coplanar with this circle, B must
be its pole. In this case the angles of the versors are right, and
FIG. 16.
the versors are unit vectors. But a glance at the figure shows
that the versor products have oppositely directed angles, and the
products are therefore unequal (compare figs. 15 and 16).
For coplanar versors, the arc AB = CD in fig. 17, and
OC OB OC OD OD OC
O A
FIG. 17.
"That the square of a right versor is equal to negative unity is
well illustrated by fig. 18, for which
\OA/ OB " OA OA lj
-the vector OB being perpendicular to A'A.
J.Q. B
18 " MULTIPLICATION OF QUATERNIONS. [CHAP. ir.
_ r Urq in fig. 15 by Up, we have the new figure (fig.
19), since vr=Upq~ l and 'Uqr 'Uqpq- 1 . The point Q is the
pole of the versor Ug or the extremity o the vector UVg.
The arcs AC and A'D are equal, and equally inclined to the
great circle ABA' since the angles of the triangles ABC and A'BC'
are equal. Thus AC may be changed into A'D by a rotation
round Q through the angle AQA', double the angle of the quater-
nion q. The vector U Vp to the pole of the arc AC is transformed
into ITVqpq- 1 by the same rotation. Now Vqpq~ l = q- Yp. q~ l
because V(q.Sp.q- l ) = Q 9 S(g.Vp-2~ 1 ) = J and accordingly a
conical rotation round the axis of q and through double its angle
changes an arbitrary vector p into the vector
1 Ex. 1. If OP is the vector from a fixed point to a point in a rigid bo'dy,
rotation of the body round an axis OQ=V^ through an angle %Lq carries
the point p to P', where o?'q . OP . q' 1 .
< Ex. 2. The displacement produced by the rotation is
PP'^.OP.^-OP.
* Ex. 3. A translation of ? the body carries a point from P to p", where
pp"=S is the same for all points of the body.
Ex, 4. If the body is first rotated, as in Example 1, and then translated
the displacement of p is
while if it is first translated and then rotated, the displacement is
* Ex, 5. If the body is first rotated about one axis OQ and then about
another OR, o*'=rq . op . q-^^rq . OP . (rq)~ l .
Ex. 6. If the first rotation is now reversed, the position of the point p'
is p", where op"=q- l rq /t>P . q~ l r^q.
Ex. 7. A body receives rotations about two intersecting axes. Prove
that the order in which these rotations are effected is of importance.
[The displacements of a point are
gr.op.r- 1 ^- 1 OP and rq . OP . q~ l r~ l - OP,
and these are generally different unless qr^rq, but then the quaternions are
coplanar and the rotations take place about one and the same axis.]
ART. 22.] CONICAL EOTATIONS. 19
Ex. 8. Find the reflection of a body in a plane mirror.
[The point o being on the mirror, which is perpendicular to A., the vector
X . OP . X" 1 is the result of rotating OP through two right angles round the
normal. Ee versing the direction of this vector, the vector to the image of
the point p is OP' = - X . OP . Ar 1 .]
Ex. 9. Successive reflection in two mirrors is equivalent to a rotation
round the line of intersection of the mirrors through double the angle
between the mirrors.
[Here -//.(-A.op. A~ 1 )ju,- 1 = +/A.OP. Ar 1 /*- 1 . Also
where is the angle between the mirrors, and 2.LjaA.=20.]
Ex. 10. Given three lines intersecting in a point, it is required to draw
three planes, each through one of the lines, so that the lines of intersection
in one plane may be equally inclined to the contained line.
When is the problem indeterminate ?
[Let a, /3, y be the vectors of the given lines. The sought lines of inter-
section are V/?ay, Vy/3a, Yay/3. Compare Art. 31, p. 31.]
ART. 22. The laws of combination of the five symbols
S, V, K, T and U
may be summarized in the symbolical multiplication table :
S Y K T U
S
V
K
T
U
S S T SU
V -V VU
S -Y IT KU
S TY T T
TJY UK U
to be read from the left. For example, the tensor of the vector
of a quaternion is TYg; the scalar of the vector is 0; the
tensor of the scalar is Sg according as Sq is positive or
negative. A positive scalar may be regarded as the quotient of
two vectors having the same direction; for a negative scalar
the directions are opposite. Hence we may write US^ = 1
according as Sg is positive or negative. The versor of a zero
quaternion must be regarded as arbitrary, unless we know a
law according to which the quaternion diminished indefinitely.
TU#==l==UTg for all quaternions. The versor of the conjugate
and the conjugate of the versor of a quaternion are easily seen
to be equal to one another and to the reciprocal of the versor.
The symbols T and U are not distributive like the symbols S, K
and "v . Apart from change of sign, it is easy to see that the
only new combination arising from further repetition of the
symbols is TVUg ( =sin Lq).
It is necessary to make some convention concerning the notation
to be employed when we wish to denote for example the square
of the scalar of a quaternion q or the scalar of the square of the
20 MULTIPLICATION OF QUATERNIONS. [CHAP. n.
quaternion. There can be no mistake if we employ brackets and
write (Sg) 2 for the square of the scalar and S(g 2 ) for the scalar
of the square, and whenever there is the least fear of confusion
brackets should be used. One of the great advantages of
quaternions is the extreme brevity of the notation. Another
and still greater advantage is its great explicitness, and this
should never be sacrificed for the sake of a few brackets.
Hamilton writes S . g 2 for the scalar of the square and Sg 2 for
the square of the scalar whenever there is no fear of confusion,
and he uses the notation V. q 2 and Vg 2 in a similar sense and in
conformity with the established notation d . x 2 and d# 2 for the
differential of a? 2 and for the square of the differential of x.
Some eminent authorities, Tait for instance, in conformity with
the notation cos 2 x = (cos #) 2 , write S 2 g instead of Sg 2 , though in
strictness this would mean S . Sq ( = Sq). But considering the
enormous care Hamilton took with his notation we prefer to
abide by his convention. No confusion can arise with respect
to T.g 2 or Tg 2 or (Tg) 2 , for the tensor of the square is the
square of the tensor, and similarly U.g 2 = Ug 2 ==(Ug') 2 and
K.j 2 =(Kg') 2 =Kg 2 . The expression Sp ,q means the product of
Sp into g, and it is well when possible to write this in the
equivalent form qSp, while S .pq is the scalar of the product pq,
but if the expressions are at all complicated, it is safer to write
(Sp)q and S(pq).
An imaginary quaternion
where p and q are real quaternions and where \/^T is the imaginary symbol
of algebra regarded as a scalar commutative with all quaternions, is called a
Uquaternion by Hamilton. Similarly he calls imaginary vectors (a + v^T . (3)
bivectors and imaginary scalars, biscalars. No ambiguity attaches to
S=S^W^TS2, or to Vg-VjpW^TVft
and the only ambiguity in T is one of sign, and this Hamilton removes as
follows. He writes
where x and y are real scalars and where x is positive, and in order to
determine x and y he employs the relation (Art. 17}
or
observing that qKp^K.pKq, so that the imaginary part of (T) 2 may be
written ^T^lS.pK^ or 2V^1S . qKp.
Equating reals and imaginaries we find, from
aP-^Tp-Tf and
that the real positive value of x is
ART. 22.] THE SYMBOLS S, V, Z, T, U. 21
It may happen that T(QQ f ) is -TT' instead of +TQTQ' where Q and Q'
are biquaternions. In other particulars ambiguity does not arise.
The tensor of a biquaternion may vanish, and in this case we have an
equation such as
W-o,
where Q'=KQ without having either Q or Q' zero. The conditions are
T 2 =T 2 and
and when these are satisfied, the biquaternion Q is called by Hamilton a
nullifier. A few examples will be found in Chap. IV.; and the Lectures
on Quaternions (Arts. 669-675), from which this account of biquaternions has
been taken, may be consulted with advantage.*
Ex. 1. Prove that combinations of the symbols prefixed to q lead to one
or other of the following :
S?, Y?,.K, T?, U0; TV?, SU$ YU?, TYU?, (Uq)-\ TTVq.
Ex. 2. Express these functions in terms of #, #, z, w, i, j and Jc. (See
Ex. 1, Art. 17, p. 13.)
Ex. 3. Express these functions in terms of the tensor, axis and angle of
the quaternion.
* Ex. 4. Show that the vectors UYpg and UY . 'UpUq are identical.
Ex. 5. If a, B and y are vectors, prove that Y is a redundant symbol in
S.aV.jSy.
' Ex. 6. Find the difference of the expressions S . pqr and S . pV . qr.
' Ex. 7. If TTVp = VUp, prove that Sp = 0.
* Ex. 8. What inference can be drawn from the equation Y^=YU^ ? and
what from "Vq=~Uq ?
Ex. 9. Prove that
T(y+j8)>(Ty-T0) unless Uy=-U/3,
and find the relation in the exceptional case.
1 Ex. 10. Show that
Tq+Tp>T(q+p) unless q=x
Ex. 11. Show that
T^ + SgOO unless <q=ir.
EXAMPLES TO CHAPTEE II.
' Ex. 1. Prove that Y(a-/3)(a+/3)=2Va/3 and assign the geometrical
interpretation.
- Ex. 2. Show similarly that S (a - /3) (a + ft) = a 2 - /3 2 and interpret.
. Ex. 3. Under what conditions is (a-f/5)(a -ft) equal to a 2 - ft 2 ?
* Clifford uses the word biquateriiion in another sense, and Prof. A. M'Aulay
has rechristened Clifford's biquaternions, and has written a large book entitled
"Ocbonions: a Development of Clifford's Biquaternions." (Cambridge, 1898.)
It does not seem to be unreasonable to retain Hamilton's convenient word for the
purpose for which it was coined.
22 MULTIPLICATION OF QUATERNIONS [CHAP. n.
Ex. 4. Establish the identity connecting three quaternions,
p*+q*+r*=pqr+qrp+rpq, where
* Ex, 5. If the relation
1 I =
13 a a/
connects two vectors a and ft, pr6ve that aft-^ar 1 ^^ 1 and show that the
vectors are parallel.
j * Ex. 6. Reduce any two quaternions p and q to quotients of vectors
| having a common denominator, or in other words, find three vectors a, /3
5 and y, so that
p = P q=I.
r a' * a
* Ex. 7. Prove that the relations
p+q~3, -<7=- Z where >=-, <?=-,
^ * a ' ^ * a J ^ a' * a
are consistent with the definition that the sum of quaternions is the sum of
their scalar parts plus the sum of their vector parts.
1 Ex. 8, For any two quaternions
' q(q~ l r~ l ) = (r q) r" 1 ; *q(q r)~ l r = (ir l q- 1 )' 1 .
Ex. 9. The sign V is superfluous in S . aiVfty. Is it superfluous in
a
Ex. 10. The second vector a may be omitted from Va(a+#). Mav it
be omitted in Ver l (a + j8) or in Va(a+ / 8)- 1 ?
Ex. 11. Contrast, where necessary, the four expressions,
Ex. 12. The laws of refraction of light from a medium of index n into
one of index ri are comprised in the relation
where v, a and of are unit vectors along the normal, the incident and the
refracted ray, respectively.
(a) From this relation,
n'a! = T\/(?i' 2 + ^Vvcr) -
Ex. 13. It is required to find a quaternion q and vectors a, /3 and y so
that if a, 6 and c are three given quaternions,
<7=<x, bq=6. cgy.
(a) Show that * ^ ' 7
a__a 6 fi c y
5-J8 1 c~y ? a = a J
and explain how a, ^ and y can be found from these relations; the tensor
of one vector (a) being assumed. (Robert Russell)
CHAPTER III.
FORMULAE AND INTERPRETATIONS DEPENDING ON
PRODUCTS OF VECTORS.
ART. 23. It is often useful to consider a vector as representing
a directed area. Assuming any two vectors a, $, so that Va/3
may equal a given vector y, we may regard y as representing the
directed area of the parallelogram determined by a and /3 there
being as many units of area in the parallelogram as there are
units of length in y. The shape of the area represented by a
vector is arbitrary as well as its position; its magnitude and
aspect are determinate. For there is obviously no reason why
this representation should be confined to the areas of parallelo-
grams.
Ex. A force is represented in magnitude and line of action by the line AB.
The moment of the force at the point o is represented by
Y . OA . AB.
ART. 24. The scalar of the product of three vectors is the I
volume of the parallelepiped having conterminous edges equal to \
the vectors.
The transformation
S . a/3y=S . a(V/3y+S/3y) = S . aV/3y
shows that this scalar is equal to the scalar of the binary product
of a into V/3y that is, it is the negative product of the projection
of a on the normal UV/3y to one face into the area of that face.
If rotation round a from /3 towards y is positive, the volume is
Sa/3y, for the angle between a and UV/3y is then acute, and
SaUV/3y is negative.
Ex. 1. If Sa/3y=0 the vectors are coplanar, and conversely.
Ex, 2. Prove that interchange of any two vectors changes the sign of
Ex. 3. Prove that
Sa/3y=Saa'a" if /3=#a+a', y=ya+za'+ a".
FORMULAE AND INTERPKETATIONS. [CHAP. in.
Ex. 4. Prove the identity,
* Ex. 5. Prove that SAB . AC . AD is six times the volume of the tetra-
hedron ABCD.
AET. 25. The formula
V.aV / 8y==ySa j 8-- j 8Sya . .................... (l.)
is very important owing to its frequent occurrence. Since the
vector on the left is perpendicular to V/Sy it must be coplanar
with $ and y that is, it must be of the form x/3+yy where x
and y are scalars. But the vector is also perpendicular to a.
Therefore Sa(cc/3+2/y) = 0, so that the ratio of x to y is
determined; and the vector must be parallel to
w(/3Say ySa/3).
It remains to determine w to satisfy
V . aV/3y = w(/3Say-ySa/3).
Multiply by ya and take the scalar part of the product, and we
have
S. yaV.aV j 8y = ^Sya j 8Say = Sya(ay/3y-SaV J 8y)= -SyaSa^y,
so that w= 1.
The proof here given is merely illustrative of a general method.
Hamilton's proof is as follows. Since '
2V . aV/3y = aV/3y- V/3y . a= a(/3y-S/3y)- (/3y-S/3y)a
on adding the pair of cancelling terms /Say /3ay, we have
2V. aV / 8y = (a j 8+ / 8a)y-^(ya+ay) = 2ySa y 8-2/3Say.
Adding aSjSy to each side of the formula, we find the relation
Y. q j 8y = aS/3y-/3Sya + ySa ) Q, ................. (II.)
which is occasionally useful.
' Ex. 1. Prove that j^ . ^
V . Vaj8Vy8ttSj878-)8Soy8=8Saj87-ySaj88. '
Ex. 2. Prove that S . Va/?VyS = SaSS^y - Say S/3S.
[ThisisS.aVjgVyS.] "
Ex. 3. Find the direction of the common edge of the planes parallel to
a and /? and to y and 8.
[The normals to the planes are parallel to Va/3 and VyS.]
>Ex.l Prove that S.V/3yyyaVa^=-(Sa/3y) 2 .
ABT. 26. The formula
pSaffy = aSffyp + /3Syap + ySaffp ............... (i.)
is of great importance, as it enables us to resolve a vector along
three vectors a, j8 and y which are not all in the same plane. It
is virtually proved in Ex. 1 of the last article. ( h ~ V ; *
ART. 28.] RESOLUTION OF VECTORS. 25
Otherwise assume, as we may, provided Sa/3y is not zero,
and operate by S/3y (that is, multiply by /3y and take the scalar
of the product). This gives S/3y/o=ccS<x/3y.
Another valuable formula is
VfiySayo + VyaSflo + Vgggy^ ............ (n.)
which enables us to resolve a vector p into components at right
angles to the planes of a/3, /3y, and of ya. Assuming
and operating by Sa, S/3 and Sy, the unknowns x, y and z are
found.
Ex. 1. Prove that ttS/3yp+,#Syap-hyStt/3p==0 if Say=0.
[Here aa+6/3+cy = 0, where a, 6, c are scalars. Operate by Vo, V/?
and Vy in turn, and we find Y/3y : a=Vya : bVa/B : c.]
Ex. 2. In the same case, V/3ySap+VyaS/3p+Va/3Syp==0.
- Ex. 3. Eliminate p between the equations
Sap-1, S/3p = l, Syp = l, SSp = l.
Ex. 4. Eliminate the scalars x and y from the relation
ART. 27. To resolve a vector along and perpendicular to a
given vector, observe that
/^x.x-^-xsx-^+xvx-v ................... (i.)
In case the essentials of a problem turn on two vectors a and
/3, put X = Va/3, and the transformation
p = Va^(?ap)- l p + cSp(VaP)- l p-&a<yap)-*p ....(n.)
will often be found useful. (Compare Art. 25.)
An expression of an analogous type is
Spa/3-aS/3p+/3Sqp
'
ART. 28. The squared tensor of jS a is
................... (l.)
for CS
Hence for a plane triangle
The identities Vaj8=Va(j8-a)V^O-a)
lead to the remaining fundamental formulae of a plane triangle,
^ sin A sinB sinC
26 FORMULAE AND INTERPRETATIONS. [CHAP. in.
* Ex. 1. If T(p~a)=T(p-f a), prove that Sap=0.
Ex. 2. The equations
; T/>=Ta
a p
are consequences one of another.
EXAMPLES TO CHAPTER III.
* Ex. 1. If V.g-a^O, where q is a real quaternion and a a real vector,
show that
S2=0, Vq\\a.
Ex. 2. The relation V.<?a=V.a'g implies a 2 a' 2 , and S.fa-a
where Sg does not vanish. It may be written in the form
* Ex. 3. Provided S# is not zero, the relations of = gag* 1 and V . qa= V . afq
are equivalent.
Ex. 4. If a!=qaq-\ the quaternion q is expressible in the form
~ a+a"
where x and y are arbitrary scalars.
Ex. 5. The same quaternion may also be written '
q = u + y ( a 4. a') + ?0Vaa',
provided a single relation connects u 9 v and w. Find it.
Ex. 6. If a'=qaq- 1 and /B'=q/3q-\ show that to a scalar factor
Verify that this agrees with the expression given in the last example.
Ex. 7. If three vectors a', /?, y' are derived by a conical rotation from
three others, a, /3 and y, prove that it is possible to determine scalars #,
y and , so that
o?(a / -o)+y()8'-j8)+*(7'-y)0.
Ex. 8. If a, ^ and y are any three vectors, and if q is any quaternion
we shall have
S . qaq- l /3y +S . ^/fy^ya-f S . gyqr^ft = S .
Ex. 9. If three vectors satisfy the relation
(a^=_a^V,
they are mutually at right angles. If they satisfy
(a/Jy^+a^V,
they are coplanar.
Ex. 10. Given that Va^S=0, prove that the four vectors are coplanar,
and show that the condition is equivalent to
u r u f-
Interpret this result.
Ex. 11. In any product of coplanar vectors a 1 a 2 a 3 a 4 ... a*, it is allowable
to transpose among themselves in any way the vectors with even suffixes
and also to transpose the vectors with odd suffixes among themselves
CHAPTEE IV.
APPLICATIONS TO PLANE AND SPHEEICAL TRIGONOMETRY.
Coplanar Versors.
ART. 29. In dealing with rotations in a plane, let i be a unit-
vector perpendicular to the plane, and let angles be measured in
the sense of positive rotation round i. If
17(7 = cos A+i sin A, .......................... (i.)
the versor Uq has its angle equal to A, provided A is less than
two right angles, and generally whatever magnitude the angle A
may have, Lq = A+mx where m is an integer. Hamilton calls
A the amplitude of the versor U#, the new name being intro-
duced to avoid any confusion as to what is meant by the angle
of a versor. (Compare Art. 17, p. 13.)
It follows from the laws of multiplication of quaternions
{Art. 21, p. 17) that
= <
if Ug = cosA+isinA, Ur = cosB + tsinB,J ........... ^
provided A and B are less than two right angles, and this result
evidently remains true when A and B are any angles whatever.
But in full, since f = 1,
Uq . Ur = (cos A + 1 sin A) (cos B + 1 sin B) \(rn }
= cos A cos B - sin A sin B + i(sin A cos B + cos Asin B),J ^
and therefore on comparison with (IL), since 'U(qr)^Uq . Ur, we
obtain the formulae for the expansion of eos(A+B) and of
sin(A+B) on equating separately the scalar and the vector parts.
The angle of (Uq) n is n times that of Ug, provided n is an
integer and TI /-<?<>; and generally when n is an integer, the
amplitude of (Uq) n is n times that of Uq. If the amplitude of
Ur is one ra 1 * that of Ug, and if the two versprs are coplanar, Ur
is one of the rrfl* roots of Uq ; or we may write
(iv.)
28 APPLICATIONS TO TEIG-ONOMETEY. [CHAP. iv.
- n
More generally the amplitude of (Uq) m is that of Uq, and in
f ffb
a similar manner we can interpret the expression (Uq) x , where x
is any scalar, as a versor coplanar with U#, and having its ampli-
tude # times that of U#. If A is the amplitude of Uj, we may
write
2A
Ug = *', .................................. (v.)
2A
for the amplitude of Ug is times a right angle, and the ampli-
tude of i is a right angle ; and still more generally, any quaternion
may be expressed as a power of a vector,
= ct, where a = UVj . Tj 51 *, t
Concerning the n ih roots of a quaternion q which are coplanar with it, it
must suffice to remark that these are n* in number, being the solutions of
the equations,
(vn.)
n i n. Ti-
* y
TTT3
if qa+cb and V$'= : ^+^) since a
so that in addition to the n real quaternion roots whose amplitudes are
!/ !, _r 2?r T , 2(n-l)w x
n"*' n L *+^ "^ + - L ir i -' .................... (VI11 -)
there are ?i(^- 1) imaginary quaternion roots corresponding to the imaginary
solutions of the equations (viz.).
The exponential e, where q is a quaternion, is defined by the formula,
and because quaternion multiplication is not commutative,
is
unless # ha/ppens to be coplanar with p. In general, however, because S<7
V, q and K? are commutative in order of multiplication,
tf^&.e 7 *, e K 2=e s ^- v ?, e^e^
and also by the definition of e it follows that
and thus T^=^, U^=:<3 v ff=cosTy ? + Uy^sinTV^ ................. (XL)
substitution in (ix.) and separation of the scalar and vector parts affording by
the known formulae for the expansion of a sine or cosine the second expres-
sion for U^. r
ART. so.] POWERS OF QUATERNIONS. 29
If we write <?=log<?'j "where ^^L^gf^ ........................ ( XIL )
we liave by (xi.), S ] og gf = log T#', V log q' = log U^' ;
and generally if p and q are any two quaternions, we may define
ffl = #l*P 9 ..................................... (Kill.)
but as we shall not require much, or indeed any, acquaintance with the
logarithm or exponential of a quaternion in the sequel, we refer to Hamilton's
Elements of Quaternions for further details.
Ex. 1. Prove that a+/W-l is a square root of zero, where Ttt=T/?,
Sa/3=0.
[See Art 67, Ex. 1.]
Ex. 2. Show that a product pq may be zero without having p or q equal
zero.
[If pq is a scalar, q must be proportional to "Kp. The squared tensor of
\/-Tp 2 +p is zero. (Art. 22, p. 21.)]
Ex. 3. Show that a quaternion q satisfies an equation of the form
<2' 2 + 2#-l-y=0 when x and y are certain scalars.
Spherical Trigonometry.
ART. 30. If a, /S and y are three coinitial and unit vectors
determining a spherical triangle ABC, the whole doctrine of
the spherical triangle is contained in the relation
FIG. 20.
The vectors
' = UVa/3,
terminate at the vertices of the polar triangle, rotation round
these points from A to B, from B to C and from C to A being
positive; and in terms of these vectors the equation may be
written in the forms,
fi .- = K^; (cosc+y / sinc)(cos6+j8 / sin&)=cosa a'sina. (IL)
Observing that rotation round OA from C' to B' is negative,
the versor
y'/3' = COS(TT B'C') a sin(7r B'C 7 ) = cos A a sin A,
30 * APPLICATIONS TO TRIGONOMETBY. [CHAP. iv.
and thus on expansion of (IL), we have
cos c cos b + y' sin c cos b + f sin b cos c + sin b sin c cos A
a sin A sin b sin c=cos a a sin a ................ (in.)
The scalar part of this equation gives the fundamental relation
cos& = cos& cos c+sin b sinccos A; ............... (iv.)
while the vector part is
a sin A sin b sinc = a / sin a+(3' sin b cos c+y sin c cos 6. ...(v.)
Operating by Sa on this vector,
sin A sin b sin c = sin aSdUV/3y = Sa/3y ,
so that =55 = = ____r_._- ............. (vi.)
sin a sin 6 sine sm a sm b sin c
Now (compare Art. 17 and Art. 25),
l=T(a/3y) 2
= (Sa/3y ) 2 - (Va/3y) 2 - (Sa/3y) 2 - (aS/3y - /3Sya + ySa) 2
= (Sa j Sy) 2 -(a 2 S/3y 2 +/3 2 Sya 2 + y 2 Sa/3 2 -2S/3ySyaSa/3),
and accordingly, in terms of the sides of the triangle,
Sa/3y =s + (1 cos 2 a cos 2 6 cos 2 c + 2 cos a cos b cos c)*, . . .( vn.)
and thus the remaining fundamental relations are established.
2x 2B 2c
Ex. 1. Prove that a*/3*y" = - 1,
rotation round a from B to y being supposed positive.
fy r f /?' St 1
[For the supplemental triangle ^r- ^ % 1, ~^~ ==< /~ v > etc - (compare
Art. 29 (v.)).] 7
Ex. 2. Deduce the relations
cos c + cos A cos B = cos c sin A sin B,
y sin c = a sin A cos B + ft sin B cos A + Va/5 sin A sin B.
Ex. 3. If P is any point on the surface of the sphere and Q the foot of
the perpendicular let fall from this point on the side AB, prove that
cos PC sin c = cos PA sin A cos B + cos PB sin B cos A + sin PQ sin c sin A sin B.
Ex. 4, Taking P at the centre of the circumscribing small circle, prove
2 cot R sin 2 == sin A sin B sin c,
where E is the radius of the small circle and where 2 is the spherical excess.
Ex. 5. Show how to represent versors and their products by versor
angles analogous to the versor arcs of Art. 21. p. 16.
2L(r-C| ^ 1 B ^
[By Ex. 1, y* ^^/J*, so that if the versor a' is represented by a
2B
directed angle A at the extremity of the vector a, and if ft" is similarly
ART. S].] FUNDAMENTAL FORMULAE. 31
represented by a directed angle B at the extremity of ft ; the product is
represented by the directed external angle TT-C at the extremity of y.
FIG. 21.
To construct the product of two versors p and q on this plan, let A be the
extremity of UVp, and B of UV#. Draw the great circle AB, and the great
circles AC and BC making the angles Lp and Lq with AB, and intersecting in
the point c, round which rotation from A to B is positive. Then pq is
represented by the external angle at c. To construct the product qp, a
point c, must be similarly found below AB, so that rotation round it from
B to A is positive. The method may be extended to spherical polygons
(Elements of Quaternions, Art. 313).
ART. 31. In his fifth and sixth lectures and in Art. 297 of the
Elements of Quaternions, Hamilton has developed at consider-
able length a curious and interesting theory connected with the
"fourth proportional" /S/x," 1 -)// to three given vectors and with
the area of a spherical triangle ABC, whose sides are bisected in
A,, B, and C, by the extremities of these vectors.
The vectors a, /3, and y terminating at the vertices of ABC, and
A,, B y , C, being the middle points of the sides of the triangle, we
have the relations,
32 APPLICATIONS TO TRIGONOMETRY. [CHAP. iv.
and from these relations or directly, we find
y-a^Sa,- 1 , a-Ay/8,- 1 , jS-y/iy,- 1 ................ (H.)
Hence a =
(m)
if
is the "fourth proportional" to ft /t a, and y,, so that the conical
rotation produced by p( )p~ l leaves the vector a unchanged, and
therefore a is the axis of the quaternion p.
Again we have
so that the conical rotation in question produces the same effect
on the vector y, as the conical rotation round P the pole of the
great circle A;B, through. twice the angle of fta," 1 . And because
the point C, can be converted into the extremity of py,p~ l by a
rotation round P or round A, this extremity must be the reflection
of C, with respect to the great circle PA. Thus the angle of the
quaternion p is C,AL if + a is its axis, while it is C y AP if a is
its axis, and we proceed to show that the former alternative
is true.
The point P being the pole of A;B,, the angles L and M are
right. Taking ON perpendicular to A X B, it follows that the
triangle NCB, is equal to LAB, and that NCA, is equal to MBA /?
for NOB, has the side B,C, the angle CB,N and the right angle
CNB, equal respectively to the side AB,, the angle AB,L and the
right angle ALB, of the triangle ALB,. Hence AL is equal to BM,
both being equal to ON ; the triangle APB is isosceles, its equal
sides being complements of AL or BM; and the equal external
angles C X AL or C 7 BM of this triangle are equal to (A+B + C),
C,AL+C;BM being A+B + B,AL + A;BM=A + B + B / CN + A / CN.
Moreover, if we join PC,, the angle PC,A will be right, C x being
the middle of the base of the isosceles triangle APB; and the
angle C,PA will be equal to z./3/z,' 1 , for it is z.BPA or JML or
A y B x , since by the equality of the small triangles MA 7 = A X N and
NB X = B X L. Hence by the construction of Ex. 5, Art. 30, the angle
c;PA represents /3/x," 1 and AC X P represents y y , so that C,AL repre-
sents p or /3 / a / ~ 1 y / , and therefore
a .............. (IV.)
Again we have this remarkable transformation by (L),
ART. si.] SPHEEICAL EXCESS. 33
so that for the new quaternion,
a, .............. (VII.)
if 2 is the spherical excess of the triangle ABC, because
EXAMPLES TO CHAPTEK IV.
Ex. 1. If a is a unit vector at right angles to (3, show that
aj8 = a-,
where w is a scalar.
Ex. 2. If a, /3 and y are unit vectors, mutually at right angles,
Ex. 3. Given two sets a, /5, y and a', /?', y' of mutually rectangular unit
vectors in the same order of rotation, so that a'=+j@'y' if a=+/3y, show
that we may connect the two sets by the series of relations
(2) &=&, yg^yiCOstf+aiSintf, 03= -y^in #+04 cos 6 ;
(3) y'=y 2 , a / = a 2 cos<+/:?2 sin <, (3'= -ctgsin^+^cos^ ;
and draw a figure to exhibit the Eulerian angles \^ 5 and <.
Ex, 4. The conical rotation <?( )q~ l which converts the first set of vectors
of the last example into the second is determined by the versor
(see Tait's Qvatemiotis, Art. 373); while other expressions for the same
versorare Sfc S * S * * i
v-tty'ffi'yl'-fy'PF'y** and z-y^V-
Ex. 5. Given in order n coinitial vectors a L , cu, ... a^ it is required to
draw n planes, each through one of the vectors, so that the lines^of intersec-
tion of each plane with the two adjacent may be equally inclined to the
contained vector. Prove that the vector along the intersection of the planes
through a x and a n is parallel to Va u a w _i ... ai.
Ex. 6. Show that
where a, /? and y are any three unit vectors.
J.Q. C
34 APPLICATIONS TO TBIGONOMETJRY. [CHAP. iv.
Ex. 7. If a, @, y and 8 are the vectors from the centre to four points
A, B, c and D on a sphere of unit radius, show that
ZA 2J3 2 2Z>
a'jS'y'S'-l,
when the quadrilateral is uncrossed, and when rotation round an internal
point from A to B to c to D is positive.
(a) Hence
(cos A -f a sin A) (cos B + @ sin B) = (cos D - 8 sin D) (cos c - y sin c).
(b) Also
cos A cos B sin A sin B cos AB=COS D cos c - sin D sin c cos CD ;
and if P is any fifth point on the sphere from which perpendiculars PQ and
PR are let fall on the arcs AB and CD,
sin A cos B cos AP + cos A sin B cos BP + sin A sin B sin AB sin PQ
-f sin c cos D cos CP + cos c cos D cos DP 4- sin c sin D sin CD sin PR = 0.
(c) Examine the cases in which P is taken to be the pole of a side or of a
diagonal, or the point of intersection of AB and CD. (See Elements of
Quaternions, Art. 313.)
Ex. 8. If a'=UVy, /?=TJVya, /=UVa& where Ta=T^=T 7 = l,
and Sa/3y <0, prove that aUV/3y, j8=UV/a' and y-UVa'/S'.
(a) If A, B and c are the supplements of the angles between the pairs of
vectors /?, y' ; y', a' ; and a', /3', deduce the relation
2^ 2J3 20
a' r /3' r y' r = -l.
(b) Show that this equation may be transformed into
**.e*0.0y=-l.
(c) Examine whether it may be further simplified to
^a+JS/3-Ky^-l,
and carefully state your reason. (Bishop Law's Premium, 1898.)
CHAPTER V.
GEOMETBY OF THE STRAIGHT LINE AND PLANE.
ART. 32. The vector p = OP being drawn from a fixed origin
and being regarded as variable, the equations
0, and V/>/3 = 0, ........................ (l.)
represent respectively the plane through the origin perpendicular
to a and the line through the origin parallel to /3.
If y = OC, <J=OD, the equations or a plane through C and a
line through D are respectively
S(/>-y)a = 0, and V(p-)j8 = ................ (IL)
These may be replaced by
, and p = 6+/3t, ................... (ill.)
where T is an arbitrary vector subject to the single implied
condition Sar=0, and where t is an arbitrary scalar.
The point E in which the line intersects the plane is the
extremity of the vector,
The first of these expressions has been found by substituting
S+{3t for p in the first equation (n.) of the plane. The second
has been found by replacing p by y+ar in the first equation of
the line. Another expression for the vector to the same point
of intersection is
From (iv.) we have the intercept DE = e S on the line, and
the interval CE = e y in the plane between the fixed points
and the point of intersection.
36 THE STRAIGHT LINE AND PLANE. [CHAP. v.
If we make in (iv.) } /3 = a, we find the foot of the perpendicular
from the point D on the plane to be at the extremity of the vector
OM = / a = ^-a- 1 S((S-y)a or /x = y + a' 1 V(y-<S)a J ...(VI.)
since the line being now parallel to a is perpendicular to the
plane.
The vector perpendicular from the point D on the plane is
(VII.)
and it will be noticed that we may directly obtain the vectors
DM and CM by resolving the vector DC along and perpendicularly
the vector a. (Art. 27.)
If in (iv.) we replace a by /3, we find the foot of the perpen-
dicular from the point C on the line to be the extremity of
the vector
or v = 7 +/3~ l V(y-S)fr (vm.)
because now the plane is perpendicular to the line. The vector
perpendicular is
CN=/3- 1 V(y-^ = /3- 1 V / 8CD ................. (ix.)
In general the normal to the plane (11.) makes with the line an
angle determined by
or sn = or
a a
and if we are required to draw a plane through the point C
making a given angle with the line, we have
U/3=cos0Ua+sin0Ura; while Ua = cos0U/3+sin0Ur/3,...(xi.)
if the line is to be drawn inclined at a given angle to the plane.
In these equations the vector T is arbitrary, subject to the implied
conditions, which are Sra = and Sr/3 = respectively,
Ex. 1. Two objects, B and c, are observed from the origin of the vector a
to be in the directions 11/3 and tly, and from the extremity of a to be in the
directions \J^ and Uy' ; prove that the vector BC is
VaU
and point out the conditions implied in this expression.
[For the point B we have xU/3=a+yU/3 f , and therefore
Ex. 2. Four points A, B, c, D are viewed from a fifth point P. Prove that
they appear to form a parallelogram ABCD if
U(UPA+UPC)=U(UPB+UPD) ;
a rectangle if UPA + UPC = UPB + UPD ;
and a square if in addition STJPA . PB = SUPB . PC.
ART. 33.] PEKPENDICULABS. 37
[The first condition requires the diagonals AC and BD to appear to bisect
one another. The second requires that they should also appear to be equal,
and the third imposes the additional condition that adjacent sides should
appear to be equ "
Ex. 3. Find the equation of the locus of a point equidistant (1) from two
fixed points, (2) from two fixed planes.
Ex. 4. The extremity of the vector p is projected from the extremity of
the vector a into a point on the plane SA/> -1-1=0. Prove that this point
lies at the extremity of the vector
SA(a-p)
ART. 33. The equation of a plane through the points C, C', and
of a line through D, D', are respectively,
S(/>-y)(y'-y)a = and V( P -(5)0$'~<5) = 0; ........ (i.)
or S( p y + yy'+ya = and V(p<$+ AT +<$ = (): ...... (n.)
,
and =
the plane being determined by the condition that the vectors OP
and CO' shall be coplanar with some fixed vector a, and the line
requiring that DP shall be parallel to DD'.
The various expressions given in the last article may be modi-
fied to suit the present case by replacing a and /3 by V(y' y)a
and S' 8 respectively.
The plane through CC' parallel to the line DD' is
S(p-y)(v'-y)(a'-*)-0, .................... (iv.)
because the normal to the plane must be perpendicular to the
line, so that SV(y'-y) .(<$'-<$) = (), Q? a = x(y'-y)+y($ -S),
where x and y are certain scalars which disappear on substituting
in (L).
If a plane can be drawn through CO' perpendicular to DD', the
equation VV(y'--y)a.(<r-<$)==(), requiring S(y'-y)(<$'-o) = 0,
must be satisfied.
We may, without loss of generality, take a to be perpendicular
to CO', and as it easily appears that the plane for which in addi-
tion Sa(5' 5) = is most inclined to the given line, we can verify
that the minimum value of
TV. V(y-y)a.(S'-S) . Q __ S(y' -
S.V(y'-y)a.(<$'-<5) " tan ^
where the vector a is regarded as variable, and that the plane
SV( /0 -y)(y'-y)V( y '-y)(<$'-<S) = ............. (VI.)
is most inclined to the given line.
38 THE STRAIGHT LINE AND PLANE. [CHAP. v.
ART. 34. The equation of a plane through three given points,
AjB ' c>is J^(r7?^$ji^^ ...................... 00
for the condition that PA, PB and PC should be coplanar reduces
to -this expression; and in this equation V(/3y+ya + a/3) repre-
sents double the vector area of the face ABCywhile Sa/3y is the
volume of the parallelepiped having three conterminous sides,
OA, OB, 00 (Art. 24). The equation may be taken as asserting
that if through the boundary of a vector area determined by
V(/3y +ya+a/3) we draw vectors equal and parallel to OP (P being
any point in the plane), the volume of the solid thus constructed
is equal to that of the parallelepiped (Art. 23).
Writing for brevity, the equation of a plane in the form
1 ' ................................. (IL)
the vectors
M = < 5~\- 1 (S\(S-l) = X- 1 VX(J+\- 1 , and DM^X^-X^SXcS (ill.)
are respectively the vector to the foot of the perpendicular from
a point D on the plane, and the vector-perpendicular from the
same point.
To find a plane equally inclined to three given lines OA, OB and
OC, we have
cos 6 . TX = - SXUa = - SXU/3 = - SXUy,
so that (Art. 26)
UX . sec 6 . SUa/3y = - V(U/3y +Uya+Ua/3),
and the equation of the plane is
Sp V(U/3y + Uya + Ua/3) = const,
or
A plane equally inclined to the faces of the pyramid OABO is
represented by
a plane cutting off equal areas on its faces is
Sp(U V/3y + U Vya + U Va/3) = const,
while the equations of the planes cutting off equal intercepts
from the edges and from the normals to the faces have been
already found.
Ex. 1. Find a plane equally inclined to the bisectors of the angles of the
faces of the pyramid OABC.
ART. 36.] INTERSECTIONS OF PLANES. 39
Ex. 2. The planes through an edge and through the bisector of the angle
of the opposite face intersect in a line.
Ex. 3. Find the equation of the plane bisecting the angle between a pair
of faces.
Ex. 4. Find the equation of a plane through an edge and normal to the
opposite face, and prove that three such planes intersect in a line.
ART. 35. The line of intersection of the planes
SX P = 1, S MP = 1 is VpVX/i-Ai-X, or ^
and that of the planes
is VpVXM=/<, or >
Three planes S\p = l, S/x/> = m, Svp = n intersect in the point
pSX^ = V(J//+mi/X+7iX/A); .................. (IL)
and the condition that the planes should intersect in a line is
V(lpv+mv\+n\p) = Q, ..................... (in.)
if I, m and n are not all zero. If they are all zero, the condition
is SX/Ai/=0 ............................... (iv.)
Four planes intersect in a point if the condition
S(lfjLv& mXi/or+TiX/xsy pXjuv) = ................ (v.)
is satisfied, the equation of the fourth plane being S/o57=p.
The conditions of intersection (m.) and (v.) may be replaced
by the pairs of simultaneous equations
and x\+yim+zv+wTx~Q, xl+ym+zn+iup=zQ ....... (vn.)
respectively, the compatibility of the equations (vi.) or (vn.)
being equivalent to (ni.) or (v.).
ART. 36. Given a pair of lines
V(p-y)u = 0, or p = y +ta\ and V(p-y')a /=s0 or p = 7 '+1fa', (l.)
the vector from a point P on the first to a point P' on the second is
PP^y-y+tfV-te .................. . ...... (n.)
If it is possible to select the scalars t and If so that this vector
may vanish, the lines intersect and the condition of their inter-
section is
S.PPTa / a = 0; or S(y'-y)aa = 0, ............. (m.)
P and P' being arbitrary points on the lines.
40 THE STRAIGHT LINE AND PLANE. [CHAP, v,
Resolving the vector PP' into two components, parallel and
perpendicular to the vector Wa, which is at right angles to the
directions of the two lines,
= a
Vaa Vaa
and substituting from (n.) on the right,
(IV.)
Thus the line joining the arbitrary points has a fixed com-
ponent perpendicular to the directions of the two lines, and
suitably selecting the scalars t and if in (iv.) we see that
are respectively, the vector-perpendicular to the two lines, or the
vector shortest distance from the first line to the second, and
the vectors from the origin to the feet of this shortest vector the
points P and P '.
Ex. 1. Verify that P O P O '= OP O ' - OP O in equation (v.).
Ex. 2. Draw a line through a point (E) to intersect two given lines
V(p- r )a=0, V(p-/X=0.
[The line is parallel to V . V(c - y) aV(e - y') a'. See (m.).]
Ex. 3. The locus of a line which intersects three given lines is repre-
sented by
(a) Reduce this equation to the form XYZW, where X, Y 9 Z and W
are planes.
Ex.4. "Writing o-=Yp 1 p 2 , T=p 2 -p 1 ,
prove that <r and r are merely multiplied by a scalar, if for pj and p 2 are
substituted the vectors to any two points on the line of their extremities.
(a) Conversely, given' any two vectors, cr and r, satisfying the relation
So*T=0, show how they determine a line parallel to T.
(6) In this notation any two lines may be denoted by the symbols (o-, T)
and (cr', T'). Prove that the lines intersect if
SoV+So-V-0.
(c) Any scalar relation homogeneous in the pair of vectors cr and r
imposes a single condition on a, line.
ART. 37.] ANHARMONICS. 41
(d) If the planes SA 1 p + l=0, SXo_p+ 1=0 contain the extremities of the
vectors p 1 and /> 2 , show that
where 2 is some scalar.
(e) Hence any relation homogeneous in the pair of vectors cr and r when
equated to zero may be expressed in the forms
/(<r, r)=0,
(/) According as the equation /(a-, r)~0
is equivalent to one, two or three scalar equations, it represents a complex,
a congruence or a regulus of right lines, and the constituents of the vectors
cr and T, when resolved along three mutually rectangular directions, are
Pliicker's coordinates of a line. (See Salmon, Geome&y of Three Dimensions,
Chap, xin., Section n.)
(g) The lines of a complex /for) = ( f being now a scalar function), which
pass through a point, the extremity of the fixed vector p 1? generate a cone
/(V /3l r,r) = 0;
and the lines which lie in a fixed plane, SXjp+1 =0, envelope the cone whose
vertex is the origin and which is the reciprocal of the cone
/(or, -VX 1 cr)=0.
ART. 37. The vector to any point on the line joining two
given points A and B is
t being a variable scalar. If P x and P 2 are any two points on
the line, their vector distance is
and the anharmonic ratio of any four collinear points is
/PPPP \ _ P i p 2 P 3 P 4 _ (*a "" ^i) (^4 "" ^3) / TTT \
v^i-^a^s^ p^p" pp /* __/ w/ M ............ ^ "'
-^2^- r ^ r r
In particular
More generally, the anharmonic ratio of any four points
I> Q2> Qs an< ^ 04 collinear with any two points P', P", of the range,
The two ranges (i.) and (v.) are homographic.
Ex. 1. If the range APBP' is harmonic, prove that
JL_ 1 = _2_ or 1 1 = 2
AP AP' AB' p-a p'-ci fi-a
42 THE STKAIGHT LINE AND PLANE. [CHAP. v.
Ex. 2. Any two homographic ranges situated on a common line,
aa + tb/3 ,_cy+td
p - a + tb ' p ~"c+td'
may be simultaneously reduced to the forms,
Ex. 3. Show that the vectors and rj satisfy the equation,
ART. 38. In many problems relating to a tetrahedron, it is convenient to
have the equations expressed in a symmetrical manner, and some of the
following relations will be found occasionally useful.
If the vectors X, //., v and tTT are the vector areas of the faces of a tetra-
hedron ABCD we may write
These vectors are independent of the origin, and their sum is zero, or
3X=X+^+v + ^=0 ................................. (n.)
Again, if , m, n, p are the sextupled volumes of the pyramids subtended
at the origin by the four faces,
Z-S/JyS, m=-Saya, tt=Sa/2S, ?=-Sa/?y; ............. (m.)
and their sum is the sextupled volume of the tetrahedron, or
?,l = l+m+n+p v, ................................ (iv.)
and is independent of the origin. Also,
^la = la+m]3+ny+p8z=Q .............................. (v.)
Changing the origin to the extremity of the vector co, and putting
a'=a-oj, etc., the volumes subtended by the faces at the new origin are
l'= S^yS'= S( - u))(y - o))(S - o>), etc.,
or I'l SoA, *m f = m So>/x, ?&'=?& -So>i/, p'=p SwST. ......... (vi.)
But still (by v.),
and this reduces by former results to the new relation,
a)^+^aSa>X = 0, ................................. (vn.)
which holds for all vectors co. Operating on this by So/, we may write the
result in the form, So>(w'2^+SASao> / )=0 ; and, because o> is arbitrary, the
part within brackets must vanish. But co' is also arbitral^, and accordingly,
for all vectors oj, we have
coSZ + SAScoa-O ................................. (vm.)
Again, it is easy to see that
2aX=aA + ^+yv4-S^=-3-y=2Aa; ................... (ix.)
and, for verification, it is sufficient to take the terms in a/3y, which are
aV/?y - /?Vcoy +y Va= - 3p.
The sum 2aA. is independent of the origin.
On the whole, we have
. ...(x.)
ART. 40,] TETEAHEDEON. 43
It is sometimes convenient to employ the vector perpendiculars from the
vertices on the opposite faces instead of the vector areas. If a /5 y, and 8,
are these vectors, it is easily seen that
v=a,X.=frp,= 7/ v=8fi ) .............................. (XL)
because, in fact, the equation of the face BCD may be written
S/>A=Z, or S(/>-u)A=y, or SOo-aJa,- 1 ^!.
Thus (x.) gives
S-=0, 2Z=w; 2Ja=0; ~cotf=2-Sa>a==:2aS- ; -3=2-=2-a....(xn.)
a / a, a, a, a,
Ex. 1. Prove that the vector sides of the tetrahedron are given in terms
of the vector areas of the faces by the relations
= -08-8)0;
and show how to connect the rule of signs with that for the expansion of
a determinant of the fourth order.
Ex. 2. Show that
Ex. 3. Given the magnitudes of the areas of the faces of a tetrahedron,
show that the directions of the normals UA, U/t, and Uv to three of the
faces must satisfy the relation
TW = TA 2 + Tju, 2 + Tv 2 - ST/zvSUjui/ - 2Ti/ ASU v A - 2T A/xSU A/A.
ART. 39. Any five vectors are connected by relations of the form
aa+fy8H-ey+$J+e=0, where a+b+c+d+e=Q ; ............ (i.)
and if the vectors are drawn from a common origin o, and terminate at the
five points A, B, c, D, E,
a : b : c : d : e=(scDE) : - (ACDE) : (ABDE) : - (ABCE) : (ABCD), ......... (n.)
where (ABCD) is the volume of the tetrahedron determined by the four points
A, B, c, D.
To prove this, remark that if
the ratios of the four scalars a, 6, c and d have the values defined by equation
(IL). (Compare Art. 24, Ex. 5.) The fifth scalar e is -(a+b+e + d).
It should be noticed that the five scalars are absolutely independent of
the origin of vectors.
Ex. Any five quaternions are connected by a relation of the form
.vp +i/q + sr + ws + vt =
where #, #, z, 10 and v are scalars.
c ART. 40. Hamilton has elaborated a remarkable system of coordinates
which he terms " Anharmonic Coordinates," the nature of which we proceed
to explain.
In accordance with the last Article we may write any vector OP in terms
of the vectors to four points A, B, c, D in the form
_ _ xcta, + yb/3 + zcy + wd8 ...... / x \
~~ "~
44 THE STRAIGHT LINE AND PLANE. [CHAP. v.
where a, 6, c and d are arbitrarily assumed constants and where sc, y, z and
w are the anharmonic coordinates'in question.
The point u at the extremity of the vector
is called the unit point, its anharmonic coordinates being equal to unity.
The point p,, whose coordinates are #+&?', y+ty'^z + tz', w + tw', is collinear
with the points p and p', for
OP
'
And, in particular, the planes GDP and CDU cut the edge AB in the points
determined by
t v
19 = - ----> 19 = - -7, ....................... (iv.)
*- sca+yb ' *- a + b ' v y
for P 12 , P and P^ are collinear, and also u 12 , u and 1134, where
cy + d8
* - 7~
c + d
Denoting by (CD.APBU) the anharmonic ratio of the pencil of planes
through the edge CD and the points A, P, B and u, we have
(v.)
and similarly, (AC.BPDF)=-, etc.
The ratios consequently of paii^ of the coordinates, #, f y, z, w of a point P
are expressible as anharmonic ratios ; and the coordinates are unchanged by
any linear transfownation, it being understood that the unit point undergoes
the same transformation as the vertices of the tetrahedron.
To suit special circumstances, the unit point may be specially selected.
It may, for example, be taken at the mean point of the tetrahedron, and
then a=b=c=d=l.
Ex. 1. The vector p of any point P of space may, in indefinitely many
ways, be expressed under the form
OP= _
P~
where aa + fy8+ey+rf8+60, a+b+c+d+e = Q.
[In terms of the four vectors a, /?, y, 8, the anharmonic coordinates of
the point are x~v,y-'V,z-v and 10 - v. See also Art. 39.]
Ex. 2. The equation of a plane ,in anharmonic coordinates being
Iv + my + nz +pw = 0,
prove that the ratios of the coordinates of the plane , m, n,p are expressible
as anharmonic ratios.
I The line AB cuts the plane in the point OL 12 = <ma ~'f/? and the anhar-
"- m -i am bl
monic ratio (AU 12 BL 12 )= - j.
Ex. 3. Find the condition that the planes I, m, n, p and ?', m' n' p
should be parallel. ^ ' 7 ^
[The plane at infinity is
ART. 40.] ANHARMONIC COORDINATES. 45
EXAMPLES TO CHAPTER V.
Ex. 1. The equation of the plane through the origin perpendicular to the
vector a may be written in any one of the seven forms,
l; =; +K=0;
p-a ' a 2 ' a a '
T(p + a)=T(p-a) ; Spa=0.
Ex. 2. The equation
T(p-a)=T(p-/3)
represents the plane bisecting at right angles the line AB.
, Ex. 3. The equations
represent respectively the half -line through the origin, having the direction
of the vector a, the half -line having the direction of - a, and the whole line
parallel to a.
Ex. 4. The equations
represent the two sheets of the cone of revolution, with o for vertex, OA for
axis, and passing through the point B (Elements, Aj?t. 196 (4)).
Ex. 5. The equation TV^=TV^
a a
represents the right circular cylinder, of which OA is the axis and B a point.
Ex. 6. If A, B, c and D are the vertices of a regular tetrahedron having its
centre at the origin, a+ +7 + s =0 ;
a 2 = 2 = etc. == - 3Sa/J = - 3S/3y = etc. ;
Ex. 7. Find the area of a face of the regular tetrahedron and the volume
in terms of the vector from the centre to a vertex.
Ex.8. The six vectors 0, )8, y terminate at the vertices of a
regular octahedron. Find the conditions the vectors must satisfy, and deter-
mine the volume, area of face, length of side.
Ex. 9. If A, B, c, D are any four points in a plane, the vectors a, j8, y,
drawn from an arbitrary origin to terminate at these points, are conne
by a relation of the form,
aa+bp+cy+dS=Q, where
/ v m^ / /
(a) The vector oc = 7 =
terminates at the point of intersection of AB and CD.
(6) If A' and B' are points similarly constructed on the remaining sides BO
and CA of the triangle ABC,
46 THE STRAIGHT LINE AND PLANE. [CHAP. v.
(c) Hence deduce the equation of six segments,
AC/ BA/ OB[ _
C'B " A'C * B'A ~~
(d) The right line B'C' meets BC in the point A", where
OA"=a'
(e) Hence A' and A" are harmonic conjugates to B and c.
(/) The equation of the six segments made by the transversal C'B'A" is
AC/ CB^ BA*_
C'B * B'A * A"C~~
(g) The points A", B", c" are collinear, and the vectors a", fl' and y" are
connected by a relation,
M'+mff'+ny", where
(h) The line AD meets B'C" in the point A"', where
,_ ,,,
OA ~ a
a-d
(i) The points B'", c'", A" lie on the polar line of the point A with respect
to the triangle BCD.
Ex. 10. Let ABCD be any tetrahedron^ and E any arbitrary point, the
vectors from an arbitrary origin to the five points A, B, c, D, E are con-
nected by the relation,
(a) The line AE meets the opposite face in A', where
OJL _ a ___
a+e b+c+d
(b) The line A'B' intersects the line AB in the point,
aa-b/3
a-b
(c) The six points formed in this way form a complete quadrilateral.
The vector to any point in the plane of this quadrilateral is of the
(e) The line AE meets this plane in the point A,, where
_4aa+e e
' 4a+e '
Ex. 11. The tetrahedra whose vertices are at the extremities of the
vectors a, fa y, 8 and act, 6/3, oy, dS respectively are in perspective.
(a) Corresponding edges intersect in points at the extremities of vectors
of the type, oa(l-6)-ffi(l-q)
a-b
(b) The six points thus determined form a complete quadrilateral.
(c) Prove that the equation of the plane of perspective may be written in
the form,
the determinant law of signs being obeyed.
ART. 40.] EXAMPLES. 47
Ex. 12. Determine a parallelepiped, having its vertices on the four lines.
joining the origin to the points A, B, c and D, and having its centre at the
origin.
(a) If -._
x+y+z+w '
a parallelepiped having its centre at P and its vertices on the lines PA, PB, pc r
PD, has its vertices at the extremities of the vectors,
-p\ pz(y-p\ pw(S-p).
(b) If a pair of edges are at right angles, the condition may be written in
either of the forms, S y =Sa ^ or * + yi aBa * + 8*
where, for brevity, a'=#(a-/>), etc.
(c) The locus of a point p satisfying this condition is a quartic surface.
(d) If two pairs of edges are at right angles, the conditions may be
written as a /2 =/3' 2 , y /2 =8' 2 .
(e) If the parallepiped is rectangular, the conditions are
tt'2^/3'2^'2^2.
(/) The point, or points, satisfying these conditions are also given by
TJ(a-p)TJ08-p)TT(y-p)TI(8-p)0,
and it may be shown that this is the condition that
T (a-p)TG8-p)T( y -p)T(8-p)
should be a minimum.
(g) Another form of this condition is
STJ.(p-/3)(p- r )(p-8)=SU.(p-a)(p- 7 )(p-S)
= SU.(p-a)(p- j 8)(p-S)
= STJ.(p-a)(p-/3)(p-),).
Ex. 13. Find the vector to a point p at which the faces of a tetrahedron
subtend volumes whose ratios are given.
Ex. 14. Find a vector equation for determining a point P at which the-
faces of a tetrahedron subtend solid angles whose sines are in a given ratio.
Ex. 15. What is the condition in terms of the lengths of the sides of a
tetrahedron that two opposite edges should be at right angles to one another?
(a) If two pairs of opposite edges are at right angles, the third pair is also
at right angles.
Ex. 16. The vectors a, ft and y are coinitial. It is required to draw
through the extremity of a a plane which shall cut the vectors in points
forming a triangle of given species. Show that the problem may be reduced
to finding scalars y and 2, so that
where Z, m and n are given scalars ; and eliminate either y or z, so as to
obtain an equation in the uneliminated scalar.
Ex. 17, If the perpendiculars from the vertices of the tetrahedron ABCD-
intersect, and if the origin is at the points of intersection, show that
48 THE STRAIGHT LINE AND PLANE. [CHAP. v.
Ex. 18. Given three points A, B, c, show that the three equations
represent a line which is the locus of the fourth vertex D of a tetrahedron
ABCD enjoying the property that perpendiculars from the vertices on the
opposite faces concur.
(a) Show that the point in which the line meets the plane of the triangle
ABC is the extremity of the vector,
__ Saffy - aSaQS - 7) " 8)8(7 - a) - yS? (a - ft)
H - 7? - V '
and express this vector in the form,
x+y+s
(&) Show that the line may "be represented by
Ex. 19. When the vector to a point P in the plane of ABC is expressed in
Deform, =
p
show that the ratios of #, //, and 2 are the ratios of the triangles PBC, PGA, PAB.
(a) Hence, if upper and lower signs correspond,
~ TG8-7)T(7-a)T(a-)
are the vectors to the centres of the inscribed and escribed circles of the
triangle.
(6) Deduce the corresponding theorem for a tetrahedron, and find the
vectors to the centres of the inscribed and escribed spheres.
Ex. 20. Selecting any point u in the plane of three given points A, B, c,
so that aa+b3+cy
-~
where a, 6, c are constant scalars ; the vector to any variable point in the
plane may be represented by
==
xa+yb+zc '
jc, y and z being the anharmonic coordinates of the point P.
(a) If a +y a +2^-2y2r 2ap-2y==0, the locus of P is a conic touching the
sides of the triangle ABC in points which connect through u to the opposite
vertices.
(6) If ys+ sa?+ 07^=0, the locus of p is a conic circumscribing ABC, and the
tangents at the vertices intersect the opposite sides in points on the polar of
u with respect to the triangle ABC, or with respect to either conic.
(c) The two conies have double contact, the polar of u being the chord of
-contact, and the anharmonic coordinates of the points of contact being
1, <o, o> 2 and 1, <o 2 , o> where o> is an algebraic imaginary cube root of unity.
(d) Given three scalars, u, v and w, discuss the arrangement of the six
points whose anharmonic coordinates are equal to these scalars taken in
different orders. Show that the six points lie on a conic. Examine the
three cases in which permutation of the scalars determines less than six
points.
CHAPTER VI.
THE SPHERE.
ART. 41. The equation
or / 5
requires the variable point P to remain at a constant distance a
from a fixed point E, and consequently represents a sphere of
radius a and of centre E.
The right line p = /3+ta meets the sphere in the points deter-
mined by the values of t which satisfy
T(/3-e+fa) = a, or T(/3-e) 2 -a 2 -2*S(/3-e)a+m 2 ==0; (n.)
and the product of the intercepts between the point B and the
sphere is independent of a, being
y 2 Ta 2 =T(/3-e) 2 -a 2 , ........................ (DDL)
while the sum of the intercepts is
6)Ua, ..................... (iv.)
if ^ and t z are the roots of the quadratic (n.).
The square of the chord cut off by the sphere is
)Ua 2 J ................ (v.)
remembering that (SXyu) 2 +T(VXM) 2 = Txy (Art. 17), and accord-
ingly the line meets the sphere in real points, only if
TV(/3-e)Ua<a, ......................... (VL)
that is, if the perpendicular from the point E on the line is less
or equal to the radius of the sphere. For contact,
TY(/3-e)a = aTa; and TV(/3-6)(p-/3) = aT(p-/3)...(vil.)
represents the tangent cone from the point B, BP being a tangent
line. Since TVX/^TXT^, the cone is real only when T(/3-e)>a.
The locus of the centres of the chords is derived from (iv.) by
putting ^(t 1 + t 2 )a = p /3, and is given by
J.Q.
50 THE SPHEEE. [CHAP. vi.
which represents a sphere on BE as diameter. For it expresses
that the projection of BE on BP is equal to BP, so that the angle
BPE is right.
Taking the harmonic mean of the vector intercepts to be p ft,
we have by (in.) and (iv.),
is the locus of its extremity the polar plane of the point B.
AKT. 42. Any two spheres,
p 2 -2Sap + Z = 0, p 2 -2S/3p-fm = 0, ............... ( L )
intersect in the plane,
Z-m; ........................ (n.)
and if P is any point on the second sphere and P' any point in
this radical plane, the power of the first point P with respect to
the first sphere is (Art. 41 (m.)),
or twice the projection of PP' on the line of centres into the
distance between the centres.
The spheres .cut at an angle determined by
since if a and b are their radii, a 2 +6 2 2a6cos0 = T(a /3) 2 .
For further investigation, the origin should be taken at the
intersection of the line of centres with the radical plane.
A variable sphere cuts two given spheres at constant angles,
prove that it cuts an infinite number of spheres at constant
angles. Let the sphere (L), determined by & and m, be the variable
sphere, and let it cut the spheres (a, I) and (a, I') at the angles 9
and 0'. Assume that it cuts the sphere (a", I") at the angle Q".
Then the third of the equations,
Z+m-2Sa/3=2a& cos 6 ; I'+m- 2Sa'/3 = 2a'& cos & ';
Z"+m-2Sa"/3= Wb cos 0",
analogous to (iv.), must be equivalent to a linear combination of
the other two. Multiply by scalars, x, y and z\ add and
separately equate to zero the coefficients of the variables, m, {$
and 6, and
xa cos 6 + yaf cos & + xa" cos Q" = 0.
ART. 43.] COAXIAL SPHERES. 51
The first, second and third show that the sought sphere
(a", I") must be coaxial with the given spheres, and we have, in
fact, on elimination of x, y and 2,
C6"cos ff'(l - V) + a cos 6(1' - Z") + a'cos ff(l" -0 = 0.
Substituting for a" its value, *J(Ta"*+l") } the equation
becomes a quadratic, which gives two values of V for each value
of cos 0". One sphere only is cut at right angles because the
condition becomes linear in I".
Ex. Eeduce the equations of a pair of spheres to the form,
0, where Ta=l.
(a) Prove that all spheres of the family obtained by giving various
values to w in *
intersect in a common circle.
(b) Examine the condition for the reality of the circle, and show that
whether real or imaginary, it lies in a real plane.
(c) If the circle is imaginary, there are two real point spheres of the
family. Find them.
(d) The spheres of the doubly infinite family
formed by giving all possible values to the vector /?, cut the spheres of the
family (a) at right angles.
ART. 43. Given any three spheres,
p s -2Sap+Z = 0, p 2 -2S/3p+m = 0, p 2 -2Syp+7i = 0;...(l.)
the radical planes of each pair intersect in the line,
2Sap-Z = 2S/3p~m = 2Syp-7i; ............... (n.)
or /) = K^y+mVya + ^Va/3)(S^y)- 1 + ^V( J 8y + ya+a/3).(lIL)
If the origin is taken on this line, l = m=n \ and if it is taken
where the line intersects the plane of centres ABC, the equations
of the spheres may be reduced to the type,
p 2 -2S/ep + Z = 0, S/o/=0, ..................... (iv.)
the vector v being fixed, but K being susceptible of various values.
The spheres of this family (iv.) of given radius (a) have their
centres on a fixed circle,
It is easy to verify that the radical axes of every three out of
four given spheres intersect in a point. This point is the radical
52 THE SPHERE. [CHAP. vi.
centre of the four spheres, and is situated at the extremity of the
vector '
the fourth sphere being /3 2 2S<S/>+;p = 0.
It may be verified that if in this equation p and 8 are rendered
arbitrarily variable, we fall back on the radical axis of three spheres. If,
in addition, y and n are arbitrary, the same equation represents the radical
plane of two. For example, we may put S=#a+y/3+sy, where #, y and z
are arbitrary.
Ex. 1. Find the locus of the centre of a sphere cutting three spheres
orthogonally.
[Let 8 and p determine the sphere whose centre is sought, and let the
three spheres oelong to the family (TV.). The condition ^4-jt?-2S8/c=0
must be satisfied by three values of the vector /c. Hence p= I, 8 \\ v, and
the locus is the radical axis.]
Ex. 2. Find a sphere cutting four spheres orthogonally.
Ex. 3. If four spheres are mutually orthogonal, their centres determine
a tetrahedron self-conjugate to a sphere.
[Let the spheres be referred to their radical centre. The conditions are
I = Sa/3 = Say = Sa8=S/!ty = S/3S =SyS, and the centres are conjugate in pairs
to the sphere p 2 =.]
The Method of Inversion.
ART. 44. We have seen that
represents a vector having its tensor reciprocal and its direction
opposite to the tensor and the direction of the vector p (Art. 16).
Hence more generally if
7 )-i=-&.CP- 1 , ............ (i.)
P and P / are inverse points with respect to the sphere, centre
and radius R, for
The inverse of the sphere T(p a) a is
The symbol T prefixed to the scalar on the right is intended to
show that it is to be taken positively. Thus, to invert the given
sphere into a sphere of radius 6, we have
a) 2_ a 2 according as T(a-y)> or <a, ....(in.)
ART. 45.] INVEBSION. 53
or according as the centre of inversion lies outside or inside the
given sphere.
The inverse of a plane is a sphere through the centre of inver-
sion, and the inverse of a line is a circle. Thus
or
represents a circle through the point C the inverse of the line
Ex. 1. If any two vectors OA, OB have OA', OB' for their reciprocals,
then the right line A'B' is parallel to the tangent OD at the origin o, to the
circle OAB ; and the two triangles, OAB, OB' A', are inversely similar.
(Elements of Quaternions, Art. 259.)
Ex. 2. Invert the sphere, centre A and radius a, into the sphere centre
B and radius b.
and from these
r. ---.
There are two rea] positions for the centre, but there may be only one
positive value of # 2 .]
Ex. 3. Invert a system of coaxial spheres into concentric spheres.
[A system of coaxial spheres p z -2wSap + l0 inverts into a system of
spheres having their centres on the line locus,
If this_is independent of w, it is easy to see that y 2 =0, y||a, or
y = a v' 1. _ _
The centre of the inverted spheres is a\ / -l? : ^aR? : <J~^l.
Ex.4. Prove that ^^ ^ zy
_^8 + 6 +
represents a sphere through the four points A, B, c and D.
[Invert wiwi respect to the point D.]
ART. 45. The following examples relating to a sphere and a tetrahedron
are easily solved by the formulae x. or xii. of Art. 38, or by the method of
Art. 39.
Ex. 1. Determine the sphere through A, B, c and D.
[The vector K to the centre is K= -^v~ 1 2Xa 2 = - J^A.," 1 ^, and the
squared radius is R?=
Ex. 2. Given four spheres having their centres at A, B, c and D, and
their radii equal to a, 6, c, d, find their radical centre.
[If <o is the vector to the radical centre, and if A=(o> a) 2 +a 2 , we have
54 THE SPHEEE. [CHAP. VL
Ex. 3. Describe a sphere to cut four spheres orthogonally.
Ex. 4. Describe a sphere to cut four given spheres at given angles.
[Here there are four equations of the form (K - a) 2 + a 2 - 2a# cos Q + R* = 0.
Multiplying by the scalars I and the vectors A and forming the sums, the
equations,
are obtained. Substitution for K in the first gives a quadratic in R. For
the origin at the radical centre, the equations are,
Ex. 5. To invert four spheres into four others of given radii
[If of, &', c', d' are the radii which the inverted spheres are required to
have, and if the vector i terminates at the centre of inversion,
,#=0. (Ex. 2 of last Article.)
Taking the origin at the radical centre,
-,,
a
These lead to a quadratic in R 3 for each set of signs.]
Ex. 6. Find the equation of a sphere touching the four faces of a
tetrahedron.
Ex. 7. Find the condition that five points A, B, c, D, E should lie on a
sphere.
[In the notation of Art. 39, p. 43, this is aa 2 +&/3 2 +cy 2 +d8 2 +ee 2 ==0, or
OA 2 (BCDE) - OB 2 (ACDE) + OC 2 (ABDE) - OD 2 (ABCE) + OE 2 (ABCD) = 0.]
Ex. 8. If five spheres are orthogonal to a sphere, prove that
P A (BODE) - P B (ACDE) + P O (ABDE) - P D (ABCE) + p a (ABCD) = 0,
where A, B, c, D, E are centres of the spheres and where P A , p s , p c , P D , and P B
are the powers of any point with respect to the five spheres.
Ex. 9. If five spheres cut a sixth at the angles 0, &^ etc., prove that the
radius (R) of the sixth is given by the relation
SP A (BODE) = 2/22# cos #(BCDE),
P A being defined as in the last Example, and a, 6, c, d, e being the radii of
the five spheres.
Ex. 10. Find the equation of a sphere in anharmonic coordinates.
[Compare Art. 40, p. 43. The imaginary cone standing on the circle at
infinity is
T/> 2 =0, or i2
ere is i2 4- 2o#2fo?= 0.]
Ex. 11. Prove that the equation of the sphere circumscribing the
tetrahedron ABCD is in anharmonic coordinates,
ART. 46.] INSCRIBED POLYGON. 55
ART. 46. The product of the successive vector sides of a poly-
gon of odd order inscribed in a sphere is a tangential vector at
the initial point of the polygon ; and if the number of sides is
even, the product is a quaternion whose vector part is parallel to
the vector radius to the initial point.
The centre of the sphere being 0, and A 19 A 2 being successive
vertices, the isosceles triangle A 2 A X is inversely similar to
A^O, and therefore (Art. 18, p. 14),
r
Thus, if OA 1 -=a 1 , OA 2 = a 2 , etc., A 1 A 2 = y 1> A^^yg, etc.,
2 = - y 1 a 1 y l ~\ a 3 = - y 2 a 2 y 2 " 1 = + y^Wi " X y 8 " *> etc * 5
and generally, the polygon being closed so that o +1 = ai,
aiK-y^i?" 1 ' where g = yy*-i ... y s yi ........... (*)
For an odd number of sides,
2i + i9 r = or a I Sg + So 1 Vg = 0, orSg = 0, *VqjLa ...(n.)
and for an even number,
ffaj a^^O, orV. ai Vg = 0, or Tgll^ .......... (m.)
In the first case (n odd), the product is a vector, and is perpen-
dicular to a l9 or parallel to a tangent at A r In the second case
(n even), the product is a quaternion having its vector part
parallel to a r
In connection with this article and its examples, Art. 296 of
the Elements of Quaternions should be consulted.
Ex. 1. The equation of the sphere through four given points A, B, c, D
may be written in the form
Ex. 2. The normal at the point P on this sphere is parallel to
VG-a)<a
and the vector CT being variable,
is the equation of the tangent plane at p.
Ex. 3. The equation of the circle ABC is
and the tangent to the circle at the point P is
[The vector part of a product of an even number of coplanar vectors is
perpendicular to their plane, being a product of half the number of coplanar
quaternions. Therefore when the points are coplanar the expression for the
normal vector in Ex. 2 must vanish, as this vector cannot be perpendicular
to the plane. The equation is also susceptible of geometrical interpretation.]
56 THE SPHEEE. [CHAP. vi.
Ex. 4. The product of four successive vector sides of a quadrilateral
inscribed to a circle is a positive or negative scalar according as the quadri-
lateral is crossed or uncrossed.
[Use the relation U. = U~ , which asserts that the angles ABC, ADC
are equal or supplementary.]
Ex. 5. The "anharmonic function of four points in space" being denned
by the equation
AB CD
examine the nature of this quaternion when the four points are concyclic.
Ex. 6. Prove that the anharmonic functions of any four points in space
satisfy the relations
(ABCD) + (ACBD) = 1, (ABCD) . (ADCB) = 1 ;
C'D'
and that (ABCD) = -7-7*
C B
where B', c' and D' are the inverse points of B, c and D with respect to the
point A.
[Note that a- 1 -/?- 1 ^- 1 .^-")/?' 1 -]
Ex. 7. If (OABC)=-I, prove that OB-^^OA^+oc- 1 ).
Ex. 8. Inscribe a polygon to a sphere, given the directions of the sides
of the polygon.
[Here IT? is given, q denoting the quaternion in the text ; and (n.) and
(in.) show that the vector to the first corner is _L VUg', or else || VU^.]
Ex. 9. For the gauche quadrilateral OABC, which may always be con-
ceived to be inscribed in a determined sphere, we may say that the angle
of the quaternion product, /.(OA. AB.BC.CO), is equal to the angle of the
lunule, bounded by the two arcs of small circles OAB, OCB ; with the same
construction for the angle of the anharmonic- L (OABC), or ^(OA : AB. BC : co).
(Elements, Art. 296 (15).)
Ex, 10. Let ABCD be any four points in a plane or in space, connected by
four circles, each passing through three of the points ; then, not only is the
angle at A, between the arcs ABC, ADC, equal to the angle at c, between CDA
and CBA, but also it is equal to the angle at B, between the two other arcs
BCD and BAD, and to the angle at D, between the arcs DAB, DCB. (Elements,
Art. 296 (18).)
Ex. 11. The vector part of the product of four successive sides of a
gauche quadrilateral inscribed in a sphere is equal to the diameter drawn
to the initial point of the polygon, multiplied by the sextuple volume of the
pyramid, which its four points determine. (Elements, Art. 296 (43).)
ART. 47. To inscribe a polygon in a sphere so that its sides
may pass through given points.
Let the unit of length be selected equal to the radius of the
sphere. Let the centre be taken as origin, and let p, p v p& ...
p n (=p) be the vectors to the' vertices, while /3 1? /3 2 , ... j3 n are the
ART. 47.] INSCRIPTION OF POLYGON. 57
vectors to the fixed points. The rectangle under the segments of
the chords through fa is
.................... (I.)
so that A . ~=7 = -! if A = & Si- 1 .......... (*>
Again,
and it is easy to see that, in general,
Finally, p = (-) if pn = f)) Pn = p> q ^ q ............. (V-)
Two cases now arise according as n is odd or even. In the
first place, if n is odd, remembering that p 2 = 1,
', or
or, separating the scalar and the vector parts,
Spp + Sq = Q and Spq-Sp = Q ................ (VI.)
Introducing the imaginary of algebra, these may be combined
into the single relation,
S( / o + x/ :r l)(g + V~I^) = .................. (vii.)
The equations (vi.) give a line locus for p which intersects the
sphere in two points real or imaginary which satisfy the
conditions.
In the second place, if n is even,
pp-pp = pqp + q = p(qp-pq)l or V . p ~Vp = p V .
Adding to each side x = Spp, we have
Vp Vq + Yp = p " l x = xp ; and this gives SVpVq =
on operating by SVq. Hence,
as we see by adding SpVq to each side. Thus,
as appears on taking the tensor, remembering that Tp 2 = l. This
quadratic in cc 2 has one negative root. The other root is positive,
and there are thus two real values for cc, and two real points
satisfying the conditions.
58 THE SPHEBE. [CHAP. vi.
We have now to determine p and q. Multiply p m in equation
(iv.) by x/^T and add it to q m , and
This gives at once, on referring to (EL),
, ,
and the real and imaginary parts of this product are q and p.
A quaternion of the form q + \/ ~ 1 . p is called by Hamilton a
bi-quaternion. (Compare Art. 22, p. 20.)
Ex. Show that in the notation of this article
[Multiply q+\/~^lp into Kq+J -iKp and separate the real and the
imaginary parts.]
EXAMPLES TO CHAPTEK VI.
Ex. 1. The sphere which has its centre at the origin, and has the vector
OA, or a, with a length Ta=a, for one of its radii, may be represented by
any one of the following equations :
0; S^-=l
' p + a
T(p~ca)
which are transformations one of the other, and each of which exhibits some
geometrical property of the surface.
Ex. 2. The circle which has its centre at the origin, which lies in the
plane Sa/o=0, and which has Ta for its radius, is represented by the equation
Ex. 3. If t is a variable parameter, in absolute magnitude not greater
than unity, the equations
represent a system of circles which generate a sphere.
Ex. 4. The equation of the sphere through the four points o, A, B, c may
be written in the forms
S(OA.AB.BC.CP.PO) = ;
r 1 - a" 1 ) - 0.
Ex. 5. If we project the variable point p of a sphere into points A\ B\ C N
on the three given chords OA, OB, oc by three planes through that point p
parallel to the planes BOG, COA, AOB, we shall have the equation
OP 2 = OA . OA N + OB . OB N + OC . OC*
ART. tf.] EXAMPLES. 59
Ex. 6. The expression*
in which r is a given scalar, i 9 j, Jc mutually rectangular unit vectors, while
s and t are parameters, represents a sphere concentric with the origin.
The expression may also be put under the form
and it may be expanded as follows :
p = r { (i cos tir +j sin far) sin sir + Tc cos sir }.
(a) Show how to establish the first form of the expression by the
properties of conical rotations.
Ex. 7. Show that the equation
/M
in which w is a real scalar capable of receiving any value consistent with the
reality of the vector p, represents the portion of the plane S(p a)/3=0
included within the sphere T(p~a)=T/J.
Ex. 8. The equationt T(w+p)=l,
in which p is a real variable vector and w a real variable scalar, represents
the region enclosed by the sphere Tp=l.
15-g T 9. A sphere passes through the intersection of the planes SAp=0,
SjLtp=0, Svp 0, which cut off caps the sum of whose areas is equal to 2?ra 3 .
Show that the locus of the centre is represented by
3Tp 2 + Tp . S (U A. + Uju, + Uv)p = a 2 .
Ex. 10. The centre of a sphere of constant radius a describes a circle of
radius b concentric with the origin and in the plane Sap = 0, Ta=l. The
equation of the surface generated may be written
T( Z>U . a^Vap- p)=a ;
or 26TVa/> = (Tp 2 + Z> 2 - a 2 ) ;
or 4& 2 (Sap) 2 = 4& 2 Tp 2 - (Tp 2 + 6 2 - a 2 ) 2 ;
or 4a 2 Tp 2 -4& 2 (Sap) 2 =(Tp 2 -6 2 +a 2 ) 2 ;
or />-a(a?-y)l_ b.
^p + aCa'-POr^'
or p = 5U . a-^ar + allr (r a variable vector).
(a) Taking ft and y, two auxiliary unit-vectors perpendicular to one
another and to a, show that
a 2 Tp 2 - & 2 (Sap) 2 = a 2 (Syp)^ ^
and prove that each of the planes
touches the surface in two points and cuts it in a pair of circles.
* Examples 1-6 are taken from Hamilton's Elements of Quaternions.
fThis and the last example are to be found in Hamilton's Lectures on
Quaternions, Art. 679.
60 THE SPHEEE. [CHAP. vi.
Ex. 11. If p and q are variable quaternions, while a and ft are given
vectors, show that
op =p
represents the shell included between the spheres
T/>=T(Ta-T/3).
(a) If y is a third given vector, and if a and b are given scalars, the
point P terminates on the circle of intersection of the spheres
when the quaternions p and q are connected by the relation
apap~ l -f bqfiq- 1 = y .
(b) When the relation
Vy (apap- 1 + bqftq- 1 ) =
connects p and q, the locus of P is the surface
(<?) If the condition
Sy (apap- 1 + bqfiq* 1 ) =
is satisfied, the point p must render the expression
2 - 2T/> 2 a 2 ),
less than zero.
Ex. 12. The bars AB, BC and CD are connected by universal joints at
B and c, and also to two fixed points A and D. If p is a point fixed in BC
and if we write '
where u is a given scalar, and also
where a, /2, y, 8 are given vectors and p, q and r variable quaternions
prove that '
^ being a scalar, and hence show that the inequality
determines the region within which the point P must lie.
(a) If the bar BC remains parallel to the fixed vector /3, the locus of p is
the intersection of the spheres
(p- tt /3) 2 =a 2 , (p'-ttW-^.
(b) In this case the locus of the bar BC is the cylinder
(c) When the quadrilateral ABCD is coplanar and when the motion is
confined to the plane ABCD, find equations of the form
ART. 47.] EXAMPLES. 61
for the path of any point of a plane lamina attached to BC, t, being a constant
unit- vector perpendicular to the plane ABCD, and /(#, y) being a scalar
function of x and y.
Ex. 13. Solve the equation
p-a p-ft p-y p-S~~
(a) If p', a', ft' and y' are the vectors from the point D, the extremity of
the vector 8, to the inverses of the extremities of p, a, ft and y with respect
too, 111
Hence deduce the relations
p-a p- 7 -a -
(b) Solve similarly the quaternion equation
1,1 1 1
-- -- _ ---- .
q~ a qo qc qd
by assuming
q'-d) = (a-d)(a'-d)=(b-d)(b'~d)=(c-d
(Robert Eussell.)
CHAPTER VII
BIFFEEENTIATION.
ART. 48. The equation
(I.)
in which a variable vector p is given as a function of a variable
scalar t> represents a curve in space, it being possible in general
to pass from one point P to another point P' on the locus, only in
one definite way namely, through the series of points deter-
mined by the variation of the parameter from t to t'.
The chord PP' of the curve is
(n.)
and for the sake of argument we shall suppose that the para-
meter t represents the time, so that P is the position of a moving
point at the time t, and P 7 its position at the time f.
Q/
<aC
FIG. 23.
Writing Etf--l ...................... (in.)
it is apparent that had' the point passed from P to P', in the time
ART. 48.] DIFFERENTIAL OF VECTOR TO CURVE. 63
' t, not along the curve and with varying velocity, but along
the chord and with uniform velocity, and that had it continued
to move uniformly along the production of the chord, it would
have reached the point Q' in unit time. In a similar manner the
point Q" would have been reached in unit time had the point
moved uniformly along the chord PP" in the time in which it had
described the curve and had its motion been continued along the
chord without alteration. In the limit PQ represents rigorously
the velocity at the point P, in magnitude and direction, for Q is
the position the point would have reached in unit time had it
left the curve at the point P, preserving unchanged the velocity
it actually possessed at that point. The equations
'-' - (-)
are equivalent modes of expressing the limit to which we
advance ; the third being perhaps in closest agreement with the
illustration. It is usual to write
(V.)
as an abbreviation for the limit.
The vector <p'() is the derivative, the derived or the differential
coefficient of the vector function <j>(t) of the scalar t, and the
differential of $() corresponding to any scalar differential
dt of tf is . t
^(t).dt .......... (vi.)
This is a vector tangential to the curve and of length propor-
tional to the differential dt which may be large or small.
If t is the arc of the curve, the vector <j>(t) is of unit length,
for in this case we may consider t to represent the time for unit
and uniform velocity along the curve.
If 0'() = 0, the extremity of the vector OP = 0() is a cusp or
stationary point.
Ex.1. The curve p
represents an ellipse of which a and /3 are conjugate radii.
[The vector //=-= = -asinif + ^cos#=acosf^ + n+/3sin^+n is the
radius conjugate to p.]
' Ex. 2. The parallelogram determined by conjugate radii of an ellipse is
constant in area.
64 DIFFEKENTIATION. [CHAP. vn.
Ex. 3, How is the point at the extremity of the vector
related to the points t and tf on the ellipse ?
Ex. 4. The curve p=a 2 4-2/&-f y is the trajectory of a point moving
with uniform acceleration.
Ex. 5. What is the curve
Investigate its properties.
Ex. 6. A helix is represented by
p = a cos if + ft sin t + y if,
the vectors a, /3 and 7 being mutually rectangular, and the tensors of
a and ft being equal. Determine all particulars.
Ex. 7. A conic is represented by the equation
Its centre is at the extremity of the vector
[The curve meets an arbitrary plane in two points. Find the pole of a
chord, and in particular of the chord at infinity.]
Ex.8. The equation YpaV/3p = (Va/3) 2
represents a plane curve a hyperbola of which a and ft are the asymptotes.
Ex. 9. "Write the equation of the conic of Ex. 7 in a vector form
independent of the parameter.
ART. 49. A vector function of two parameters, t and u,
p = <j>(t,u\ ................................. (I.)
represents a surface. It may be regarded as generated by the
family of curves u = constant, t variable', or by the family
t= const.
In strict analogy with Art. 48, (vi.) we have
mn\ <f>(t+At, u+-du) <h(t, u) \\
L \ 77& n / 1 / \
/ > \ '/
lim r;[0(<+M*, u+g&u)-d>(t, u)]
where d^ and du are any scalars. It is evident that this expres-
sion is linear with respect to dtf and du, so that we may write
w ......... (m.)
ART. so.] VECTOR OF SURFACE. 65
The derived vectors <j>'(t, u) and <f>j(t, u) are tangential respec-
tively to the curves u= const. and = const, at the point t, u ; and
more generally the vector fidt+fadu is tangential to the
surface.
The equation of the tangent plane to the surface is
S(/>-0)0V,=0, or S(/>-0> = 0, if HIV^, ....... (iv.)
and the vector v is normal to the surface. The equation of the
normal is
> or y (/o-0)^ = ? r p = 0+#i/ ...... (v.)
Ex. 1. If <fr(t) is a function of a single parameter, the equation
p =
represents a developable surface.
[This surface is generated by the tangent lines to the curve p =<
The normal vector is Y (<j>'+uq> ").<$ or v <"<', and is independent of u.
The tangent plane is S(/t>-<#>-^') v ^"=0 or S (/>-<)<'<" =0, and as this
is independent of u> it touches the surface all along the generator determined
by t. Conceive the tangent plane to roll over the surface and the successive
generators to become attached to it, the surface will be unfolded or developed
in the moving plane.]
Ex. 2< The equation
in which a is a constant vector, represents a cylinder standing on the curve
p<p(t) and having its generators parallel to a. The equation
represents a cone standing on the same curve and having its vertex at the
extremity of a.
Ex. 3. Find the locus of a line joining corresponding points on two
horuographically divided lines AB and CD.
rrru , . a+t/3+s(ly+tm$) .. a-K/3 ly + tmS .,
[The surface is p= r-f^ V. ^ \ if P= i . V> Pim are IQ
L I + t+s(l+tm) ^ l + t ' l+tm
homographically divided lines. This is a hyperboloid of one sheet.]
Ex. 4. Show that the variable line determines homographic divisions on
the lines AC and BD.
Ex. 5. Find the scalar equation of the locus of Example 3, and show
that it may be reduced to the form
where JT, Y 9 Z and W are planes.
Ex. 6. Find the locus of
d CD.
AKT. 50. The equation
P=0(> f*, v), ............................. (L)
in which t, u and v are variable parameters, may at pleasure be
J.Q. E
Ex. 6. Find the locus of a line similarly dividing two given lines AB
and CD.
BIFFEEENTIATION. [CHAP. vn.
irded as determining (i.) a singly infinite family of surfaces,
example, the surfaces found by assigning various but constant
les to Vj (n.) a doubly infinite family of curves, for example,
triable, u and v constant ; (in.) any point in space, for we can
;eneral find one or more sets of values of t, u, v corresponding
in arbitrary vector />. The scalars t, u, v are curvilinear
rdinates of the extremity of the vector p.
Differential of a quaternion function.
LRT. 51. The differential of a quaternion function of a
ternion is defined by the equation
............ (i.)
n=co \. \ *v / J
d .7 = lu
jfinition in complete agreement with the results of Art. 48.
J tie function f(dq) is a linear and distributive function of
differential dg, while it also in general involves the quaternion
i its constitution. To prove this proposition, observe that if
id s are any two quaternions,
f s)
n
imply f(r+*)=f(r)+f(8) ......................... (n.)
^ a corollary, f(xr)=xf(r), .............................. (ill.)
is any scalar.
.s an example,
thus * * d.g 2 t=g.dg + dg.g. ...................... (iv.)
ART. 52.] QUATERNION FUNCTION. 67
There is a notable difference between the differential of a
function of a single scalar and a function of a quaternion, which
is clearly illustrated by this example. In general, from a differ-
ential of a function of a single scalar d . F(x\ we can form a
3 -Tlf \ ^
differential coefficient ^ , which is absolutely independent of
d cc 2 d tf 2
dx. Thus, -4 = 2#, but -4^- = 9'+dg . q . dg- 1 is not indepen-
dent of dq. And this, which is a consequence of the non-
commutative law of multiplication, is really quite in keeping
with the ordinary theory, for if F(x, y) is a function of two
independent scalars x and y, we cannot form a complete
o jji "^ 77"
differential coefficient from d.JF(xy)=dx+ dy, where dx
ox dy
and dy are arbitrary, though we can of course form the partial
"^ T7t f^ T7T
differential coefficients and ^ . We must remember that a
3# 3y
quaternion is a function of four numbers, and that a differential
dq is susceptible of a quadruply infinite system of values.
As a second example,
d.g- 1 =-g- 1 .dg'.g- 1 , ..................... (v.)
for d.g- 1 =lim w(gH dq] q- l \
%=oo \.\ n / j
Ex. 1. Prove that
[Note that these symbols are distributive, or that
Ex. 2. If v is a vector function of a variable vector p, and if dv=<
show that <dp is a linear and distributive vector function of $$>> so that for
any pair of vectors </> (a 4- j8) = <f> (a) 4* < (ft).
*[This is a particular case of (u.). Fuller details will be found in the
following chapter.]
ART. 52. The differential of a function F(q, r, s, ...) of any
number of quaternions is the sum of the differentials with
respect to each separately, or
d . F(q, r, s, ...) = d g . F(q, r, s, ...)+d r . F(q t r, s, ...)
68 DIFFERENTIATION. [CHAP. vn.
Inhere d q F(q, r,s>...) denotes the differential of the function on
the supposition that q alone is variable. We may write
a. *<?, *,,...)
=lim ,
and this, by the process of the last article, leads at once to (L).
Thus,
d.qq~ l = Q = dq.q- l +q.d.q- 1 , d. g- 1 = g" 1 . dq . q' 1 . .
Generally in any product of variable quaternions, the rule~ is to
differentiate each quaternion in the position it occupies^
Ex. 1. Differentiate r=aqbqc, where q is variable.
Ex. 2. Differentiate (qrf and jV, where q and r are both variable.
ART. 53. The differentials of the functions-Sg, Vg, Kg', Uq,
Tg, UVg, etc., of a quaternion are naturally of importance. We
have already stated that
dSg = Sdg ? dVg=Vdg, dKg = Kdg, ............... (i.)
and these results are immediate consequences of the distributive
character of the symbols, S, V, K.
Since (Art. 17, p. 12)
Tg 2 = qKq, we have 2Tg . dTg = dq . Kq + q . Kdq = 2SdqTLq .
(compare Ex. 6, Art. 20, p. 15), and since Kq = Tq(Uq)" l } the
differential of Tq is
Further, since
g = Tg.Ug, and dq^dTq .Uq+Tq .dUq,
we have on division by g,
dq . . .
- < m >
and therefore by (n.), =V. ................................ ( IV .)
In particular for vectors,
dT P 2 = -a.p a = -
and d p =Tp , dTIp + U/o . dTp, and therefore,
dp dUp v dp
S =V
ART. M.] FUNCTIONS OF QUATERNION.
The relations S = 0, S=0 ..................... (VL>
\JC[ Up
are worthy o notice.
r Ex. 1. Eesolve dp into components along and perpendicular to p.
2tt
' Ex. 2.' If p=ra /?, where Ta=T/3=l, Sa/3=0, and where the scalars
r and u alone vary, show that
V^-odir, 'S^-.
P P *'
(a) Prove generally that TV.dpp" 1 is the differential of the angle swept
out by the varying vector op=p.
Ex. 3. If /> and p' are inverse points, the origin being the centre of-
inversion, and if dp and d'p are any two differentials of p, and dp' and d'p'
the corresponding differentials of p', prove that
dp' , dp
=
and interpret the meaning of this relation.
Ex. 4. Compare an element of vector area with the corresponding
element into which it is changed by inversion.
[The elements are Vdpd'p and jOTDp- 4 . p^Vdpd'p . p.]
* Ex. 5. Prove that v ,
(a) aUV2=V-~Z.
Ex. 6. The vector a being constant, prove that
d . qaq- 1 = 2V . Vd^^- 1 . qaq' 1 = 2^ (V . Y^- J d^ . a) q~ l .
' Ex. 7. Prove that
da*=d#(log Ta+f Ua) a*,
where a is a constant vector and us a variable scalar ; and that
da*=#S^.a*+V^.Va*,
a a
where x is constant and a variable.
ABT. 54. If P is any scalar function of a variable vector p,
a differential of P is connected with the corresponding differential
of p by a relation of the form
dP= -8,/dp, .............................. (i.)
the vector v being a function of p but independent of dp.
70 DIFFERENTIATION. [CHAP. vn.
The rate of variation of P along any direction a (Ta = 1), may
be written in the form
d a P = -Sya, ............................ (II.)
it being understood that the suffix a attached to d signifies that
the corresponding differential of p is
dp = a ................................. (in.)
This rate of variation as expressed by (IL) is the projection of
the vector v along the vector a, and consequently the rate of
variation of P is maximum along the vector j/, being then equal
to Tj/, while it is zero along any direction normal to v.
Having given the variations of P along three non-coplanar
directions, or what is equivalent, having given the differentials
dP, d'P and d"P of P corresponding to three non-coplanar
differentials dp, d'p and d*p of p, we can determine the vector j/.
We have in fact
dP=-S*dp, d'P=-S,xl'p, d"P=~S*l"p, ....... (iv.)
and by the fundamental formula of Art. 26, p. 24, we find
Yd'pd"p -dP+ Vd>lp . d'P+ Vdpd'p . d'P ,^
"~ Sdpd'pd"p ,
Thus it appears that the vector v is derived from P by means
of the differentiating operator
d+ Vd'pdp d'+ Vdpd'p . d"
~ Sdpd'pd"p ' .........
in which dp, d'p and d"p are any three non-coplanar differentials
Of p, and in which d, d' and d" are the corresponding symbols of
differentiation. .. d> * - SP^P
-
v Ex. 1. Prove that VSa^=-a, \
- a) . T(p - a)~ 2 .
[These follow from the relation dP=-SdpVP.]
Ex. 2. Show that
* SyVS/SVSaV . Tp- 1 - - 3 .
AHT. 55. The form of the expression found in the last article
for VP suggests a new view of the subject which is applicable
in the general case when P is a vector or even a quaternion
function of p.- Suppose a parallelepiped constructed having its
edges equal to any three vectors dp, d'p and d"p, and having its
ART. 55.] THE OPERATOR V. 71
centre at the extremity of p. If we suppose the vectors arranged
in positive order of rotation (compare Art. 24), Vd'pd"p is the
outwardly directed vector area of the face having its centre at
the extremity of p + dp ; and Vd'pd"p is likewise the outwardly-
directed area of the face, centre p-Jdp. Also Sdpd'pd'p is
the volume of the parallelepiped.
Let F(p) be any function of a, scalar, vector or quaternion,
then the sum of the products 01 the outwardly directed vector
faces into the value of F(p) at their middle points is
Vd'pd>.F(p + Jdp) + Vd"pdp.
- Vd'pdV f(p ~ H>) - Vd''pd
and the quotient of this sum by the volume of the parallelepiped is
SVd'pd'p. {(JXp+jd/Q-Jk-jdp)} ,_v
-Sdpd'pd'p .......... ." A ;
Each edge being diminished in the ratio - , the quotient becomes
......... ^ *'
So that when n increases without limit, or when the parallele-
piped whose edges are -dp, - d'p, - d"p decreases without limit,
the limiting value of the quotient (ill.) is (compare Art. 51 (i.))
Sdpdpdp
Thus V *F(p) is the limit of the ratio which the sum of the
products of the outwardly directed faces of a parallelepiped into
the mean values of F(p) over the faces bears to the volume of
the parallelepiped. And the vectors dp, d 7 p, d"p being arbitrary,
the result is independent of the shape of the parallelepiped.
Take the case in which jP(p) is a vector function (a) of p, and
consider separately the scalar and the vector parts of V . <r. The
scalar part is the limit of the ratio which the sum of the scalar
products of a- into the outwardly directed elements of the sur-
face or which the sum of the inwardly directed normal com-
ponents of a- into the corresponding area* or which the surface
* Remember that Sa0 is minus the length of one vector into the projection of
the other upon it.
72 DIFFEBENTIATION. [CHAP. vn.
integral of the inward normal component o cr bears to the
volume. Thus if <r represents the flux of a fluid, SV<r is the rate
per unit volume at which the -amount of the fluid is increasing
at the point in unit time. In other words SV<r is the rate of
increase of the density at the point. If or is the velocity of a.
fluid and c the density, c<r is the flux, or the mass of the fluid
that crosses unit area normal to or in unit time, and SV . (c<r) is
"dc
the rate of increase of density at the point, or . Thus
ot
(V.)
For an incompressible fluid, c is constant and SV<r is zero.
In like manner, V . V<r is the limit of the ratio borne to the
volume by the integral over the surface of the vector product
V.Uj/.o-.d-d, where TJV is the outwardly directed unit vector
along the normal and dA the scalar element of area, or where
UvdA is the outwardly directed vector element of area.
Since it has appeared that these results are independent of
the shape of the parallelepiped, it follows that they are true for
any closed surface formed of a single sheet, and we have
where dv is an outwardly directed element of vector area of the
surface, and where v is the volume, the limit being arrived at
when the surface becomes indefinitely small.
^ MT. 56. Towards further elucidation of the operator V, con-
sider the analogous integral taken round the vector sides of a
parallelogram, having its centre at the extremity of the vector p.
FIG. 24.
Circuiting in the positive direction and forming the product of
the vector sides into the corresponding values of F(p) at their
middle points, the sum is
.
Collecting terms and dividing by the area of the parallelogram,
the result is
jdp)- F(p~$d p )}
TVdpd
AET. 56.] CIRCUIT THEOREMS. ' 7,3
Now let the parallelogram be indefinitely diminished by replacing
dp and d'p by -dp and -d'p, and we have in the limit,
_d / p.dJ T p~dp.dTp (
TVdpd'p .........................
But this is equal to
{V . vdpdxvdyrp . d . +Ydy P . d x . + Ydpd'p . d" . )} F P
Sdpd'pd"pT Ydpd'p
because V(Vdpd'p . Vd>rp)= -d'pSdpd'pd"p, etc , s <> * hat
integral is
(n,
if TJi/=UVdpd'p is the normal to the area about which the
direction of circuiting is positive.
As in the last article, we have for any plane closed curve"
without loops,
dp being now a vector element of arc of the curve and 4 being
its scalar area.
In particular for a vector function (a-) of p, we have separately
lim = V(VTL,V . <r)..
It is obvious on using the expanded form of V that we may
write
(v.)
or that we may in this relation at least treat V as a vector in
combination with other vectors, it being understood that V
operates on a- but not on UV.
This result leads us back to an interpretation of VVor
analogous to the interpretation of VP in Art. 54. We have
SU^VVcr=lim&^,- ...................... (VI.)
or the limit of the ratio which the integrated component of or
along the arc of a plane curve ( JSdpcr) bears to the area of that
curve, is equal to the component ( SUi/YV<r) of the vector VVcr
74 DIFFERENTIATION. [CHAP. vn.
along the positive normal to the plane. This is a maximum and
equal to TWo- when the plane is at right angles to UWcr; it
vanishes when the plane is parallel to that direction.
If Sd/><r is the differential of P (some scalar function of />), the
integral jSdpo- depends merely on the limits between which the
integral is taken (leaving aside cases in which singularities
occur), and is in fact P(p%)-P(pi) if the integration extends
from p l to p. 2 - For any small closed circuit therefore the integral
vanishes, the initial and final points of the path of integration
being coincident, and therefore
VV<r=0, if S<jdp=dP. :.:.:.:',...:: ...... (vn.)
Conversely, if Wcr=0, we must have So-dp the differential of
a scalar P; for in this case the integral round any small
closed circuit vanishes, or what is equivalent, the integral from
p 1 to p. 2 is equal and opposite to the integral back by another
path from p. 2 to p v or again, the integral from p. to p 2 is indepen-
dent of the path. These results will be extended to the general
case of curves which are not small. At present we remark that
VVVP = 0, or VV 2 P = 0, or V 2 P = scalar, ...... (vra.)
- if P is a scalar function of p, is involved in equation (VIL).
ART. 57. It is useful to express the operator V in various
forms. If, for example, as in Art. 50, we suppose the vector p
to be expressed in terms of three parameters u, v and w, and if
we write
dp-|.du=p 1 du, d / />=3jdv=/) a dt;, d"p = ^aiv = p 3 fav, (i.)
the symbols of differentiation d, d' and d" refer respectively to
u, v and w, so that symbolically
d=^.du, d'-^-.dv, d"=^.dta ............ (u.)
3u 3-y dw ^ '
On this understanding, equation (VL), Art. 54, becomes
If the parameters are so selected that the derived vectors
p v p> 2 and p 3 are always mutually perpendicular, the symbols V
and S in (in.) become superfluous, and the expression for V
reduces to the simple form,
ART. 57.] CANONICAL FORM OF V. 75
If the vector p is expressed in terms of the Cartesian
coordinates x, y and z, so that p-=ix+jy+kz, we have
This last form may be regarded as the canonical form of the
operator. We have, for example, when q is the operand,
and we shall write
so that in combination with its operand V acts as a vector in
combination with a quaternion.
Again if a is a constant quaternion, we have symbolically, an
operand being understood,
oy
a.V=*ai^-+aj +a&~ = V. Sa+SVa-VVYa;
ox "oy oz
and in combination with a quaternion, not the operand, V still
plays the role of a vector.
In combination with itself
and generally in all combinations V may be treated as a symbolic
vector. Of course some little care is necessary when V is ex-
jpressed in the general form, but it is precisely of the same kind
as the care required to distinguish between
(-)'-+' - KD-
76 DIFFERENTIATION. [CHAP. vn.
> Ex. 1. Show that if #= W+iX+jY+*Z,
v Ex. 2. Verify that
V.Vo-=V 2 .cr=:-^+|^+^)<r, where <r=iX+jY+kZ.
Ex. 3. Prove
VS.Tfa-X)- 1 ^ if p is not equal to A,; V 2 TVAp= -
V* log TVAp - ; V 2 /Tp = -/"Tp - 2Tp-y'Tp.
T^ i
[For example,
Ex. 4. Prove thatVxv.p^ -2A,; VWAV.P= -
Ex. 5. Show that
. q, (aV
Ex. 6. If P and are scalar functions of />, show that
V.PQ=VP.Q+VQ.P.
Ex. 7. If > and ^ are quaternion functions of p, show that
where the suffix is intended to denote that the affected symbols are not to-
be operated on by V.
Ex 8. Interpret the expressions
VVV'.P', SWV'./W,
where the accents indicate that a marked symbol is to be operated on by the-
correspondingly marked V.
[If P and Q are scalars, the first expression is V(VP)(V), or
This last expression is also true when P and Q are quaternions.]
Ex. 9. Find an expanded form for V 2 . PQ.
Ex. 10. Find the expression for V in terms of the usual r, 6 and <f>
coordinates. [Use the relation (iv.).]
Ex. 11. Show that q . V - K . VKq where V operates on q in situ.
[It is sometimes convenient to place the operator to the right of the
operand.]
Ex. 12. If fn(p) is any homogeneous function of p of the order n which
vanishes under the operation of V 2 , the function Tp~ Sw ~ 1 .f w (p) will vanish
under the same operator. , ' ^
ART. ss.] ROOT OF QUATERNION. 77
this relation in the form
accents after the operation.
Tp' m .V*f n =Q, and observing
that S/>V-/=~^/n because f n is homogeneous in p, we easily find the
equation reduces to m(2n+l+m)*= 0. This result is of importance in the
theory of spherical harmonics.]
ART. 58. Given a quaternion function p = F(q) of another
quaternion g, we have seen how to express dp in terms of dq
(Art. 51). It is a more difficult problem to express dq in terms
of dp, and we postpone the general method of solution for the
present.* However, there are a few cases in which the problem
can be solved directly, such as to find the differential of the
square root of a quaternion.
Here P^ or # 2 = #> ........................... (i.)
so that pdp+dp.p~dq .......................... (11.)
Multiply this by Kp and into p, and two relations equivalent
to (ii.) are obtained,
Kp.p.dp + Kp.dp.p^Kp.dq; p.dp .p+dp.p*=dq.p. (m.)
Adding, we have
dp . (Tpt+ZpSp+p^Kp . dq+dq .p
because p.+Kp = 2Sp, Kp.p Tp 2 ]
or 4. dp. pSp = Kp. dq+dq. p
because
and hence -- ..... : ................
As another example, under which this might have been in-
cluded, to find the differential of the 71 th root of a quaternion
(n being an integer), we have
-p = qn y q=p n , dq = dp.p n - l +p.dp.p n ~ 2 +...+p n - 1 .dp. (v.)
* Multiply dq into p and subtract the product pdq, and
dq.ppdq~dp.qqdp, or V. VdgVp = V. VdpVq. (VL)
Thus, with an indetermined scalar x,
V.VdgrVp+a? , V.VdqVp
- - =
Turning to (v.), we have on substitution from (vn.),
* See Art. 150, p. 273.
78 DIFFEBENTIATION. [CHAP. vn.
because q and p are commutative in order of multiplication,
and because ap = Kp. a, or a(8p+ Vp) = (Sp B Vp)a if SaVp = 0,
the vector V . VdgYp . (Vg)' 1 being perpendicular to Vp. Again,
since ^ and Kp are commutative in multiplication, and the
expression (vin.) reduces further to
Thus we have by (vn.) and (IX.) on elimination of
Y.YdgVff/
**" V? V 1
and the sought differential dp is expressed in terms of p, q and Aq,
ART. 59. "Writing the first differential of fq in the form
? ), .................................... (i.)
to indicate that it is a function of q and of dq, linear in the latter, the
second differential may be expressed by
where / 2 (ff, dq) is homogeneous and quadratic in d^.
A similar process holds in general, and in particular if dq is constant, so.
that d 2 #=0, cl 8 2=0, etc., we have
d-..fr-a./ m -,fo dq)=f m (q, dq) ......................... (in.)
Suppose that f(q) and its successive differentials up to the m th are finite-
for finite differentials of q, and consider the function
F(x)=f(q+xp) -/(?) -f -/ife 10 - j 2
in which x is a scalar and q and ^ are two quaternions. Differentiating
with respect to x, and leaving p and q constant, we find by the general
relation (in.),
Putting .17=0 in (iv.) and (v.), we see that F(x) and its successive
deriveds up to the order m-1 vanish when #=0, and consequently
AET. 60.] TAYLOR'S SERIES. 79
where r m is some quaternion function of #, q and p, and where by (v.)
Fxm
MtoP) ................ (VIL>
rt
icsso w je=0
By taking x small enough it is consequently possible to render r m
infinitely small in comparison with/ TO (gr,jp), or
TV
lim m/," v~0 ................................. (vm.>
*=oT/ m (2,^) \ >
Replacing xp by > in (iv.), what we have proved is that
j
where ?', is a function of q and p, which becomes evanescent in comparison
with f m (q, p) for sufficiently small tensors of p. This theorem is what
Hamilton calls " Taylor's Series adapted to quaternions."
In certain cases, for a large value of m, the term
becomes negligible, and we may write the expansion in the usual symbolic
form, -,
or more explicitly for a vector variable,
ART 60. Instead of differentiating a second time with the same char-
acteristic d, let the differential of
be taken for a new characteristic, d' corresponding to the differentials d'#
and dfdq of q and d<?. The result may be written
d'd./to-Afe; d'df)+/,fo d's, d?), ....................... (i.
where in full,
............ (IL),
Reversing the order of differentiation,
dd'./fe)-/^, dd^)+/ f fe d ? , d'g) ...................... (in.)
We shall now prove the relation
where r and 5 are any two quaternions replacing dq and dfq in the functions*
which occur in (i.) and (m.). "We have by (n.),
/afe r, *) =Bl C8 w {j r i(fi r+ S ri V"^' 5) J
and from symmetry this is equal to/ 2 (#, s, r).
80 DIFFERENTIATION. [CHAP. vn.
More generally, if by successive differentiation of a function f (q), a function
f n (q, 1'j, r 2 , ... r n ) is constructed, the order in which the quaternions r 19 r 2 , ...r a
are grouped ai/iang themselves is immaterial.
In virtue of (iv.), it appears that
dtt./(?)-dd' ./(<?)=/!(?, d'd^-dd'?); ..................... (v.)
and in general this difference vanishes if, and only if, d'dq=dd'q.
Ex. 1. If Q is a scalar function of p, and if d=Sj/d/>, dv=<d/>, show
that the function <j> is self -con jugate, or that Sa</2=S/?<a, where a and ft
are any two vectors.
[This is a particular case of (iv.). Compare Art. 51, Ex. 2, and Art. 62.]
Ex 2. If fi and v 9 are any two vector functions of the vector p ; if
dv^^dp) and dv 2 =$ 2 (dp), and if V operates on all functions of p on its
right, show that
. Sv 3 V . - Sv 2 V . S^V . = S(&v 8 - 4W V . ;
or in other words prove that the two operators produce the same effect on
any function of p.
Ex. 3. If JP, # and * are any three quantities or operators, not necessarily
commutative in order of operation or multiplication, show that
where fofcHP!?-^ [fo 01 *]=!>, ri^-^te rf-
Ex. 4. If jo and g' are any two quantities or operators, show that
e^pet^p+^+^+TfJ^+efa, where j^Hjw, ?] ;
and hence prove the equation connecting operators,
determinate
where v } and v 2 are any given functions of p, where v 3 is a d
function of p and where V operates only on functions on its right.
'ART. 61. To find a stationary value of the scalar function
(p), whenever a stationary value exists, we equate to zero the
first differential
(I.)
of /(p) for all differentials dp. This requires the vector v to be
zero, for otherwise Svdp cannot be zero for every differential dp,
and the stationary values are obtained by substituting in f(p)
the vectors p which satisfy the equation
*=0 .................................. (II.)
If the stationary value is subject to the condition
, .............................. (nr.)
where g(p) is a given scalar function of p, the differential d/>
is no longer arbitrary, and the conditions are
ART. 61.] STATIONAEY VALUES. 81
where X is a new vector function of p defined by the nature of
the function g(p). Considered geometrically the condition (m.)
requires the vector p^to terminate on a certain surface and con-
strains the differential dp to be tangential to the surface as
expressed by SXdp = 0. The function f(p) has a stationary value
if df(p) vanishes for every differential dp at right angles to X.
In other words we must have v parallel to X, or
or Vj/X=0, ...................... (v.)
where x is a scalar multiplier. The solutions of (IIL) and (v.)
afford vectors p which render /(/>) stationary in value.
Again if there are two equations of condition,
<K/>) = 0, M/>) = 0, ........................ (VI.)
the differential of dp consistent with these conditions must satisfy
dg(p) = SXdp = 0, dh(p) = S/xd/j = 0, ............ (vn.)
so that dp || VX,u, and if in addition f(p) is stationary in value so
that d/(/o) = 0, or Svd/o = 0, we must have v coplanar with X
and n, or
0, or Si/X/x = 0, ................ (vm.)
where x and y are two scalar multipliers. Here the three
vanishing scalar functions of />, g(p) = Q, h(p) = and Sj/X/x = 0,
serve to determine a certain number of vectors p as vectors to
the points of intersection of three known surfaces, and substitu-
tion of any one of these vectors in /(/>) will give a stationary
value.
For the solution of the equations, no general rule can be laid
down. Sometimes, indeed most frequently, it is more convenient
to deal with the equations (v,) and (vm.) involving x and y
rather than with the results of elimination of these scalars.
To examine the nature of the stationary values of /(p), it is
necessary to proceed to second differentials. For example when
there are two equations of condition, we have in addition to (vu.)
(compare Art. 51, Ex. 2, Art. 60, Ex. 1),
p^Q, (ix.)
where <, and <(> are two linear vector functions determined by
the functions g(p) and h(p\ and we must consider the sign of
..................... (x.)
when appropriate values of p and dp are substituted therein.
By adding the equations (ix.) multiplied by x and y to this we
have by (vm.)
&f(p) = S&p(<I>+K$,+y<i>J A P> where d/>||YA/*, ....... (XL)
J.Q. F
82 DIFFERENTIATION. [CHAP. vn.
the scalars x and y being given by (VIIL) in terms of v, \ and. M
by means of the relations V/xj/=0VX/x, VvX^yVXp, in which
we suppose the appropriate value of p to be substituted. For the
negative sign, f(p) is a maximum, while it is a minimum if the
sign is positive.
In like manner, when there is only one equation of condition,
we find
dy(p)=Sdp(0+cc0>i/>, where SXdyo = 0, i/+03\ = 0, (xn.)
and if d 2 /(/>) is 'positive for every dp perpendicular to X the
function f(p) is a minimum ; if d 2 /(p) changes sign for some
vectors dp perpendicular to X, the function is merely stationary ;
if d 2 /(p) is constantly negative for the differentials dp, the
function is, a maximum.
Ex, 1. Find the stationary values of Tp, subject to the condition,
: [Here dTp=^ -SUpdp=0, where dp satisfies S(p-a)dp=0, so that
Up Up -a, or p || a, or p=#a say, and the condition gives
(#-l) 2 cc 2 -{-a 2 =0, or x=\a1ar\
so that p=aaUot.]
> t Ex. 2. Find the stationary values of Tp when (p-a) 2 -t-a 2 =0, S/3p=0.
EXAMPLES TO CHAPTER VII.
Ex. 1. If op=p=a<& Ta=l, Sa=0, the locus of the point p ^will be
the circumference of a circle, with o for centre, and OB ( = (3) for radius, and
in a plane perpendicular to OA (a).
Ex.2. If op=p=V.a*/3, y = oc=Va/3, Ta=l, the locus of P is an
ellipse, with its centre at o, and with OB and oc for its major and minor
semiaxes.
Ex. 3. If under the same conditions as in Ex. 2,
the locus of P' is a circle with OB' and oc for two rectangular radii. The
equation of the circle may be written
p'=a</3'.
Ex. 4. If op=p=a*/3, Sa/3=0, the locus of P is a logarithmic spiral with
o for its pole.
Ex. 5. If op=p=Y . a 15 /?, the locus of p is an elliptic logarithmic spiral
a plane curve which may be projected into an ordinary logarithmic spiral.
Ex. 6. The equation
p=cta+a/3 with Sa/3=0, Ta=l,
represents a helix,' while the locus of the perpendiculars to the axis of the
helix which intersect the curve is represented by
p
where u is a variable scalar.*
* These Examples are taken from the Elements of Quaternion, Art. 314*
ART. ei.] i EXAMPLES.
Ex. 7. If we project the ellipse
on a plane at right angles to the vector A, the vectors a and will project
into the principal semiaxes of the projection provided
(a) They will project into equi-conjugate radii if
(5) If SaUX
the ellipse will project into a circleone of four, of which two are imaginary.
(c) The squared radii of the circles of projection are
the upper sign corresponding to the real circles.
Ex. 8. A circle of radius n~ ir /3 rolls on a circle of radius T/3 and
centre o, and carries with it a point P at a distance ZT/3 from its centre.
The locus of the point P is represented by
'
(a) Prove that d/>= jw(l +n)a(p - a'j8)df,
and assign the geometrical interpretation.
(6) If the variable scalar t represents the time, the equation of the hodo-
graph * is
and show that this curve may be generated by a point carried by one circle
rolling on another.
(cj Show that the condition for a cusp on the path of the point P is
and discuss fully the nature of this equation.
(d) Prove that the vector of acceleration of the point p for uniform
motion of the circle is
and determine the condition that the acceleration may momentarily vanish.
(e) The condition for an inflexion is found by expressing that Udp is
stationary or that Vdpd 2 p=0, and it may be reduced to
(/) Show that the inflexions lie on the circle
Ex. 9. Under the same conditions, what curve, or rather what system of
curves for various values of the scalar I is represented by p
* The hodogra/ph of an orbit is the locus of the extremity of a vector drawn
from a fixed point to represent the velocity of the moving body.
84
DIFFERENTIATION. [CHAP. vn.
Ex. 10. (a) If OQ=<(*) and OQ'=-^(W) are the equations of any two
curves the relation
is equivalent to a differential equation connecting the parameters so that
corresponding values of the parameters in an integral determine equal arcs
measured from fixed points on the curves.
(6) If the condition (a) is satisfied, the quaternion
is a versor which renders the tangent to the second curve at u parallel to
the tangent to the first curve at the corresponding point t.
(c) When the curves lie in a common plane, the condition (a) being still
satisfied, the equation
is" the locus of the pole of the second curve when it rolls along the first so
that points answering to corresponding values of the parameters t and u
remain in contact.
(d) The vector tangential to the roulette at the point p is
and this vector is at right angles to p - <f>(t) because the quaternion of (6) is
a versor.
(e) The equation of the normal at the point p is therefore
Ex. 11. The earth and a planet being assumed to describe circular orbits
round the sun, show that the apparent path of the planet is represented by
where c is the radius of the orbit of the planet and b that of the orbit of
the earth, where P and E are the periodic times of the planet and the earth,
where y and ft are unit vectors normal to the planes of the orbits and where
a is a unit vector directed towards a node.
(a) Show that the equation
determines the values of t corresponding to the "stationary points" at
which the motion changes from direct to retrograde or vice versa.
Ex. 12. Show that the equation
p^KVaP+utfp where Ta=l, Sa/3=0
represents a cylindroid referred to its centre, and deduce the scalar equation
Ex. 13. Describe the loci represented by the following equations ;
(i) p=aSA.TJr;
(ii)
(iii) p
ART. 6i.] EXAMPLES. 85
where a, ft, 7, A, ^ and v are given constant vectors, and when the auxiliary
variable vector r is perfectly arbitrary.
(a) What modifications must be made in your interpretations when r
remains constantly inclined to given direction ?
Ex. 14. (a) If Spdp=0, show that Tp is constant.
(b) If Vpdp=0, it follows that Up has a fixed direction.
(c) If Sodpd 2 p=0, show that TJVpdp has a fixed direction and the vectors
p are parallel to a fixed plane.
Ex. 15. Show that
(Elements of Quaternions, Art. 343 (9).)
Ex. 16. Prove the relations
U(a+/J)=Ua . (l+a-'
and find the development to the third order when T/3 is small in comparison
with Ta.
Ex. 17. Supposing the earth to describe a circular orbit round the
sun, show that the parallactic ellipse of a fixed star is represented by
^=-Y.y at ao- 1 .Uo-
where a- and y x a are the heliocentric vectors to the star and to the earth
respectively.
(a) Show also that
UVo-y.Tao-- 1 and U . o-Vcry . TaSyo- 1
are the principal vector radii of the parallactic ellipse.
Ex. 18. If v is the (scalar) velocity of light and p the velocity of the
earth in its orbit, the aberration of a star is represented by
U(vU<r+p)-Ucr.
(a) The earth's orbit being supposed circular, the aberrational ellipse is
given by
. a+1 ao- . Ucr
where u is the scalar velocity of the earth.
Ex. 19. Assuming the effect of refraction to be K times the tangent of
the zenith distance, show that a star in the direction of the unit vector or
appears to be in the direction of the vector
where k is the unit vector directed to the zenith.
Ex 20. If P is a point in a body attached at B and c by universal joints
to two bars BA and CD having fixed universal joints at A and D, show that
the motion of the point p is subject to the conditions implied in the
equations
DIFFEJUENTIATION. [CHAP. vn.
where a, y, and T? are fixed vectors and where p, q and r are variable
quaternions ; prove that the envelope of the point may be determined by
identifying the equations
and show that these conditions require the five points ABPCD to be coplanar.
Ex. 21. If Scrdp becomes the differential of a scalar function of p when
multiplied by a suitable factor, show that So-V<r=0.
Ex. 22. If dv is the directed element of a surface at the extremity of
the vector p. the element of solid angle it subtends at the origin is
SdvV.Tp-i.
Ex. 23, Show that
Ex. 24. The differential of a function of the vectors p and or, <r being a
function of p, may be written in the form *
d . P= - Sdp(V p - Vp'Scr'Vcr) . P
where V p and V<r operate respectively on p and on o- as explicitly involved
in P, and where V p ' operates on p as involved in </, the accents being
removed after the performance of the indicated operations.
(a) If P is a scalar function of p and <r, and if a- is a function of p which
renders P constant,
VpP-Vp'So'VaP^O.
(I) If the same function o- renders constant another scalar function O of
p and o-, the relation
(P,Q)=S.W <r PV <r QVV (r where (P, Q)=S(VpPV <r Q~V p QV <r P)
must be satisfied. And if o- can be derived from a scalar function of p by
the operation of V, we must have
(P,Q)=o.
(c) If Aj, ft, A., and /^ are any vector functions of p and o-, the operator
reduces to the form S(Ai 2 V P +ft s V,).
(d) If P v denotes the operator S(V P PV~ V^PV,,), we have
PvQ -- QvP=(P,Q),
thefel ^eSon Q "^ SCalar funetions ; and if E is an y third sealai ' function,
PvQv.E-Q v Pv.E=P v (Q, R)+Q V (E, P)=( P , ( Q> E))+(Q, (R, P))
does not involve the second deriveds of R.
(t) Hence (P, (Q, R))+(Q, (R) P ) )+(E) (P> Q))s0 .
and the operator (P, Q) v =P v Qv-QvPv.
ART. 61.] EXAMPLES. 87
Ex. 25. Bright curves are seen on a surface owing to light reflected by
scratches on the surface from a source at A to an eye at B. If the scratches
are represented by putting u const, in the equation of the surface
p =</>(, u\ show that the equation of the curves may be found by combining
the equation of the surface with the result of expressing that
is a minimum with respect to t.
(a) If the equation of the surface is/p=0 and if Fpu is the equation of
a family of surfaces through the scratches, the bright curves are given by
/P=O, sv/v^{TJO)-o)+uO)-j8)}a
(&) The bright lines due to the grooves made in turning a surface of
revolution (Tp=/S&p) lie on the surface
and meridian grooves on the same surface give rise to bright curves on the
surface
SVp(Up + /'Sp){U(p - a) + U(p - )}=0.
Ex. 26. The differential of T(p-a) corresponding to a given differential
of p ceases to be determinate when p comes to coincidence with a unless, we
know a law according to which p tends to coincide with a.
CHAPTER VIII.
LINEAR AND VECTOR FUNCTIONS.
ART. 62. A vector function of a vector, distributive with
respect to that vector, is called a linear vector function.
Thus if
0(a+/3) = 0a + 0/3, S0a = 0, S0/3 = 0, ............. (l.)
for all vectors a and ft, the function <j> is linear and vector. As
a corollary to the equations of definition
<f>(Ma) = x<j>a ............................. (II.)
if x is any scalar.
Given the vectors
a'=0a, /3' = 0/3, y' = <jfry, .................... (m.)
the results of operating by <f> on any three given and non-
coplanar vectors, the function is determinate ; for by (l.)
, __ a'Sffyp + /3'Syqp + y'Safip , .
- - - I
since / oSa/3y = 2aS/3y/> for any arbitrary vector p.
With a new signification of the vectors, a, $', y', a, /3, y, any
linear function may be reduced to the trinomial form,
(v.)
in which either set of vectors a', /3', y' or a, /3, y may be
arbitrarily assumed. For if we resolve <f>p along three fixed
vectors a, /S', y 7 , the coefficients in the resolution must be scalar
and distributive functions of p ; that is, they must be of the form
Sap, S/3p, Syp. If, on the other hand, we assume a, /3 and y, the
set a, f? and y follow, being 0V/3y : Sa/3y, etc.
Thus in any case, the general linear function is seen to involve
nine constants, the nine constituents of three vectors a, /3 and y,
or a, j8' and y'.
For arbitrary vectors, a and & if
a, ..... '.... .................. (VL)
ART. 63.] LINEAB VECTOB FUNCTION. 89*
the function <t> is said to be the conjugate of the function 0.
The conjugate for the trinomiai form (v.j is
(VII.)
Ex. 1. Given
show that
r Ex. 2. Show that Va/o^S is a linear vector function of p, and find its
conjugate.
Ex. 3. Is aT/o a linear vector function of p 1
ART. 63. From a geometrical point of view the equation
<r = 0/> ................................. (I-)
in which </> is a given linear and vector function, and in which
the vector p is arbitrary, establishes a linear transformation from
vectors p to vectors <r.
Equal vectors are converted by <f> into equal vectors; right
lines transform into right lines, and planes into planes, as
expressed by the relations
if p = a + /3;
if /> = a+/3+Uy ...(ll.)
consequences of the formula of definition (Art. 62 (i.)).
The plane whose equation is
S(/o a)/3y = becomes S(<r $a)</30y = 0; ...... (ill.)
and the vector area
Va/3 transforms into V0a</3; ............... (iv.)
while the volume
Sa/3y becomes S0a0/30y .................... (v.)
* Ex. 1. Verify that ,
Sa/ty
where a, /?, y and a', /3', y' are any two sets of non-coplanar vectors.
Ex. 2. Prove that
[Take a! along the edge of the planes of aft and of y8, and reduce Va/?
and Vy8 to Va'/? and Va'y', etc.]
i Ex. 3. Prove that "V<j>a<j>ft is a linear vector function of Vafi.
[This is practically included in the last example. Verify by the trinomial
form.]
90 LINEAE VECTOR FUNCTION. [CHAP. vin.
ART. 64. There is an inverse transformation which converts
vectors cr into vectors p, so that
and we propose to investigate this transformation.
Writing 0p = o-=VAyU, < (n.)
the conditions of perpendicularity of the vectors or, A and cr, ^ give
by the property of the conjugate function (Art. 62 (vi.)).
Thus the vectors 0'A and 0V are at right angles to />, and con-
sequently
^ being an auxiliary linear and vector function defined by the
equation
in which a and /3 are any arbitrary vectors. (See the last
Article and its Examples.)
To determine the value of the scalar m operate on (iv.^by
S<f>v, where v is an arbitrary vector, and we have
mSAA/ = S0'A0V0'v, ........... .......... ( VL )
because Sp<j>'v = Si/0/> = Svar =
Operating likewise on (iv.) by 0, we have
m0/> = 0>//-(r or / m<r =
and replacing cr by <pp we also find
so that we may write symbolically
m=0-\/r = -\/r0, ...................... (VII.)
with the interpretation that the effect of operating first by \/r
and then by <j> on any vector, or first by </> and then by \fr, is to
multiply that vector by the scalar m. This relation shows that
m is an invariant, or absolutely independent of any particular
set of vectors X, yot, v in (vi.), for by (v.) i/r is independent of the
vectors X and /* in (ry.). (See also Ex. 1, Art. 63.)
Thus wherever m is not zero, we can always pass from vectors
<r to vectors p by the relation
mp = \jscr, ............................ (vm.)
m being calculated by (vi.) and T/r by (v.); and it will be
observed that in the calculation of this scalar and this auxiliary
function, we only require the direct operation of the function 0'
on vectors.
ART. 65.] INVERSE TRANSFORMATION. 91
* Ex. 1. Show that the function ty transforms vector areas into. vector
areas when vectors are transformed by the function <'.
' Ex. 2. Show that volumes are altered in the ratio m : I in transformation
by the function <',
Ex. 3. Show that T// is the conjugate of ^ if ^ / V
[Expand SVySV<a</3, and prove that it is equal to
Ex. 4. Show that volumes are altered in the ratio m : I by the trans-
formation produced by <.
Ex. 5- Follow in detail the geometrical meaning of the transformation
employed in deducing
mp = T/rcr from <r = <p.
[See Art. 63 (iv.) and Art. 150.]
ART. 65. The transformation in the last article fails in one
case if m is zero. In that case the vectors cr are all coplanar,
the volume of any parallelepiped formed by them being zero
(Ex. 4, Art. 64); and because in general mp=\Js(r if <r = 0/>, in
this particular case, the function ^ destroys every vector in the
plane.
To cover this case, consider the general transformation for an
arbitrary function 0,^
<T = (0-l-c)/> = $ c /o and m c yo = -y/r c o% ................ (l.)
where c is a scalar and where m c and i/r c bear the same relation
to <j>+c that m and -^ bear to 0. It appears at once by (v.) and
(VI.), Art. 64, that
...................... (ii.)
so that if we write
we shall have
Now for any arbitrary value of the scalar c, the scalar m c is an
invariant, and therefore, separately, the coefficients in its expan-
sion m, m' and m" are invariants, or are independent of X, /*
and v.
By (i.) we have identically for all scalars c,
92 LINEAR VECTOR FUNCTION. [CHAP, vm*
or /
and therefore equating the coefficients of c on each side
it being understood that these equations denote that equal results
are obtained by operating with right or left hand numbers on an
arbitrary vector.
One of the transformations most frequently required in
quaternions is to invert a function 0+c, or to replace an
equation <r-(0+c)/> by m c/ o = \fsc<r; and in general the process,
due to Hamilton, as given in the text is the shortest and most
certain. We first calculate V(^ / +c)X(^ / + c) y w and express it in
terms of VX/x. Then we either calculate m c from (n.), or it is
sometimes better to calculate it directly from (v.), namely from
In particular
m/>=^cr, m'/> = ^p+xo% w"/> = x/>+<r if o- = 0/o; ...(vil.)
and thus the general solution of <r=<f>p is m / /o = x"+' y / r / ) ^ m ^ s
zero with the implied condition ^<r = 0; while if m = m' = 0, the
general solution is m"/> = <r+x/o with the implied conditions
^0-=0, \fsp+x(r=Q. In the first case (m = 0, m /= =l=0), the vector
p may be considered arbitrary in ^p there is in fact nothing to
determine it. But as -^ destroys every vector in the plane of the
vectors &, it is really only the component of the vector normal to
that plane that is of any account in \fsp. In the second case
(m=m / =0), similar remarks apply; the vector p is arbitrary on
the right subject to the condition ^y>+x<r=0. The function ^
may vanish identically, and this case we shall consider in Art. 66.
1 Ex. 1. Determine the functions m, ^ and X for the function <p=
Ex. 2. Find the auxiliary functions for <jf>p
[Find < c and ^ c for
Ex. 3. Solve the equations o-=VaV/?p and cr=Vap by the general
method, and directly.
Ex. 4. Express Vv and Xc' in terms of ^ c and Xc-
Ex. 5. Construct a linear vector function which renders four given.
vectors parallel to four others.
[The data are <<x || a', <j>$ || ', < r || /, ^ || 8', and the function is
where c is arbitrary.]
T. 66.] LATENT AND SYMBOLIC CUBICS. 93
Ex. 6. Prove that
[See equation (vi.). These relations are often useful.]
Ex. 7. Prove that
Ex. 8. Prove that the equation
/>=(<HO~ la > or
a being a fixed vector and a variable scalar, represents a twisted cubic.
[Show that it cuts an arbitrary plane in three points.]
Art. 66. From the equations of the last article connecting
<j>, x and \fs we deduce
and we have the corresponding equations for the conjugate <f>,
x ' = m" - tf ; <// = m' - m V + 0' 2 ; = m - m'f + m'^ 2 - /3 . (n.)
These may be proved by reflecting that
Sa0j8 = S<j>a<j>/3 = S/30 /2 a, etc. ;
so that for example
and from the third and fourth of these we have (m'' ^> / ) a= X /a
because /3 is perfectly arbitrary.
Let g v g 2 and g% be 'the roots of the scalar cubic,
so that m=flf 1 gr 2 gr 3 , ^
This scalar cubic is called the aten cubic of the function, and
its roots are the latent roots of the function 0.
We may now write the symbolic cubic (i.) satisfied by the
function in the form
and the same symbolic cubic is satisfied by 0'. Hence
whatever vectors a and ft may be ; or in other words the vector
(<j>-~9i) a is perpendicular to the vector (<t>-g^)(<i>g^f$' Tlie
vectors a and /3 being both arbitrary, it follows that one or other
of the vectors (tf-g^a or (<t>-g*)(<P~9$>fi must be-parallel to a
fixed direction.
But ($' g : )a is not generally parallel to a fixed direction
when the vector a is arbitrary, for if it were we should have
or (-
94 LINEAR YECTOK FUNCTION. [CHAP. vm.
where a and /3 are quite arbitrary; or symbolically, ' >
^-fltt+ft'-O, or ty-fi^fo-ft) 8 * ' ......... ( vn -)
utilizing (I.) and (iv.), and replacing m"-^ and m / -flr 1 m // +flr 1 2
by their values, &+& and gg* In this case which is quite
special, the symbolic cubic of the function degrades into a quad-
ratic (vii.). , , _ . *. A , *
We conclude therefore that the product of a pair of factors of
(v.) operating on an arbitrary vector reduces it to a fixed direction,
and writing
2
( vm -)
the directions of the vectors y p y 2 , y 3 are fixed and are called the
axes of the function 0.
We have by (v.),
( IX -)
and these vectors are generally distinct if the latent roots
9v 9* #3 are une <l ual > and tlie 7 are also g enera % non-coplanar.
Eesolving then any vector p along y ls y 2 and y s we have
Thus (<f>9i)p is coplanar with the pair of axes y 2 and y 3 , and
if y/ is the axis of the conjugate .function corresponding to the
root g v it follows from the equation
that the vector y/ is perpendicular to the plane of (<t>g^)p, and
in particular to the vectors y 2 and y 3 . If vectors are drawn from
the centre of a sphere along the axes of a function and of its
conjugate, the two spherical triangles the two sets of axes deter-
mine are supplemental.
Conceive the function <p to undergo continuous variation so
that two latent roots, g 2 and gf s , approach coincidence. The
corresponding axes approach and ultimately coincide, but their
plane is still determinate being perpendicular to y/. Similarly
all three axes may coincide in a line perpendicular to that in
which the three axes of the conjugate simultaneously coincide.
ABT.ee.] t ROOTS AND AXES. 95
We shall give an illustration of ,a function having three equal
roots. Let 0a = /3, 0/3 = y, 0y=0, then 2 /3 = 0, ^ 3 a = and
generally <p z p = 0, but <f?p and </> are not zero. The function is
<{>p = (/3S/3yp + ySyap) : Sa/3y , and 2 /> = yS/3y/> : Sa/3y = Vo-
A totally different class of functions is characterized by the
equivalent conditions that the axes are indeterminate or that the
function satisfies a symbolic quadratic and not a cubic (compare
(vn.)). If y 2 and y s are two different axes corresponding to the
same root g%, the function <j> e/ 2 destroys every vector in the plane
of y 2 and y 3 , and the function is of the form
<f>p =
and (0-01X0 -#2) =
The latent cubic has two roots equal to g 2 and the third equal
to g v
Finally a third class may be noticed that for which three
non-coplanar axes answer to the same root but a function of
this kind is simply a scalar constant.
In general the latent roots may all be real, or two may b6
imaginary. Corresponding to imaginary roots g^g + ^J'-^.g'
and <7 3 = # N/ Iff', the axes must be of the form y 2 = y + \/-~ 1 y
and ys^y-x/^y'. For (Q-gJ^-g^+^-gs)] is real and
must produce a real vector from a real vector ; but
is imaginary and produces an imaginary vector from a real
vector.
Ex. 1. Every function coaxial with a given function < is of the form
[If A 1? 7i 2 and A 3 are assumed to be the three roots of the function the
only disposable constants we find on operating by scty+yx+z on y 1} y 2 and
y 3) three equations which determine #, y and &J
Ex. 2. Coaxial linear functions are commutative in order of operation,
and conversely functions that are commutative are coaxial.
[The first part is easily proved on expressing an arbitrary vector in terms
of the axes. The second part is established by operating on the axes. Of
course one function may have indeterminate axes. If so, two axes of the
other must lie in their plane.]
Ex. 3. Find the latent and symbolic cubics for ty and X-
Ex. 4. The equation
Sp<//oo=--0, or Sp<fro< 2 p=>
represents the three planes through pairs of axes of <,
96 LINEAE VECTOR FUNCTION. [CHAP. vni.
Ex. 5. In general, if (< 2 +# . <+y)p=0,
sphere x and y are scalar^, and p any given vector, either p must be an axis,
and the corresponding root must satisfy the quadratic
or else p must be coplanar with a pair of axes, and the corresponding roots
must both satisfy the quadratic.
Ex. 6. Deduce the symbolic cubic from the result of replacing A., p. and
v by <j>p, <j> z p and <t>*P in the relation
ART. 67. Combining a function and its conjugate by way of
addition and subtraction we obtain two more functions,
(i.)
To justify the form attributed to the second function, observe
that 8^-^=0 ........................... (a)
whatever vector p may be.
The function <3> is said to be self-conjugate. The conjugate of
Ve/> is Ve/>, and the vector e has been called the spin- vector
-of <f>.
The axes of a self-conjugate function are mutually rect-
angular. The function being its own conjugate, each axis must
be perpendicular to the other two. The axes of a real self-
conjugate function must be real. If two are imaginary they
must be of the form y+^/^Ty' and y-VHTy' by the last
article, and the condition of perpendicularity requires
which cannot be, as y 2 and y /2 are both negative. Hence follows
the important proposition that the latent roots of a real self-
conjugate function are real.
If two roots of a real self-conjugate function are equal, it
must have indeterminate axes. For if a single axis corresponds
to the double root, it must be perpendicular to itself, and there-
fore imaginary.
Referred to the axes a self -conjugate function is of the form
................ (m.)
and the only special case is when two of the roots become equal
An arbitrary self-conjugate function involves only six con-
stants ; the three roots and three numbers to fix the directions of
tne axes.
Ex. 1. The axes of Vep are and cW^Tc" w ;here
mutually rectangulavand where T^Tc" ' '
AET. 68.] SELF-CONJUGATE FUNCTION. 97
Ex. 2. Find the self -con jugate part of the function
<p = a'Sap + ffSpp + y'Syp,
and also its spin -vector.
Ex 3. If a self -con jugate function transforms a given vector a into a
given vector a', it transforms any other vector /3(=OB) into a vector,
J'(=OB') terminating on a fixed plane.
[Here Sa/3'=Sa7?, and a, a' and /? are given.]
Ex. 4. Given that a self -conjugate function renders a parallel to a' and
/3 parallel to ft', it renders y parallel to a fixed plane.
[The conditions of self -conjugation require
Ex. 5. The axes of a function are mutually rectangular. It is self-
conjugate.
Ex. 6. Two axes of a function are at right angles. The spin- vector lies
in their plane.
[Syi7 2 =0, Sy 1 ^y 2 = = Sy 2 ^ / y 1 =Sy 2 (<j{)-2)y 1 , etc.]
Ex. 7. Prove that the quaternions
2 = ( A. .
are invariants.
[Verify that g^ = w^" 4- 2e, q. 2 =m r -
Ex. 8. If the vectors a, and y are mutually perpendicular,
Ya-^a 4-V/3- 1 </3+ Vy-^y = 0,
when < is self -con jugate.
Ex. 9. The planes containing a pair of axes of a function and the
corresponding pair of axes of its conjugate intersect in the vector (<-#)e,
where is the spin-vector and g is a latent root.
Ex. 10. The vector to the common orthocentre of the spherical triangles
determined by the axes of a function and its conjugate is
Ex. 11. The spin- vectors of coaxial functions lie in a fixed plane.
Ex. 12. In terms of the roots and axes
ART. 68. It happens not iinfrequently to be necessary to
discriminate between the parts of */>, x, and of the invariants
wliich arise from the self-conjugate part of <j> and those which
depend on e. We have
- V . VeX . $ M -
the terms - eSX$/x + eS/>t$X cancelling.
J.Q. G
98 LINEAR VECTOR FUNCTION. [CHAP. VIIL
This easily reduces to
^^p-V^-eSep ....................... (i.)
Thus the spin-vector of ^ is 3>e or fa.
Operating by <j> or $+ Ve we have
and if we notice that $V$ep = Ve^/o (Ex. 7, Art. 65), this reduces
without trouble to
or Jlf=m+Se0e ................. (IL)
where Jkf is an invariant of $. Changing into + c, and
therefore m into m+m'c+m''c 2 +c 3 , $ into $+0 and M into
rc 2 +c 3 , we see by (n.) that
m'^'-e 2 or M = m'+e 2 and that Jlf'=m" ...... (in.)
ART. 69. We shall give a few examples of the geometrical
meaning of the invariants of a linear vector function. (Art. 65
(IV.).)
(1) The invariant m" vanishes if the function <f> transforms a
pyramid into another having its edges on the corresponding
faces of the old.* If the vectors a, {3, y are along the edges of a
pyramid, and if 0a is coplanar with /? and y, 0/3 with y and ,
and 0y with a and {3, it is obvious that m" vanishes. Con-
versely if m" vanishes we can determine an infinite number of
pyramids which transform into others having their edges on the
faces of the originals. For assuming arbitrarily a and /3, the
equations S ^/3y = 0, SoflSy-0, ....................... (i.)
determine the direction of y; and the condition m'^O* requires
Saj60y=0.
(2) The invariant m' vanishes if transforms a pyramid
into another having its faces through the edges of the old. The
proof and the converse are the same as that just given.
(3) The sum of the projections of vectors transformed from
mutually rectangular unit vectors on the corresponding unit
vectors is constant :
if U/3Uy=Ua. ...(II.)
(4) The sum of the squares of vectors transformed from mutu-
ally rectangular unit vectors is constant :
if tJ/3Uy = Ua ...(ill.)
where m"(<}>'<j>) is the first invariant of the self-conjugate function
.
into P la * es ^rsecting in pairs
ART. 70.] INVARIANTS. 99
(5) The sum of the squares of the projections, on any fixed line,
of vectors transformed from mutually rectangular unit vectors
is constant. If X is the vector on which the others are projected
S(SX0Ua) 2 = 2(SUa A) 2 = T0'X 2 .......... * ..... (iv.)
(6) The sum of the squares of the projections on a plane is
constant.
Similar remarks apply to vector areas V0TJa0U/3, etc.
Ex. 1. If the sum of the square roots of the latent roots of <j> is zero,
it is possible to find an infinite number of pyramids (OABC) which convert
into others (OA'B'C'), so that intermediate pyramids (OA^CJ can be drawn
having their three edges in the faces of the first, while their faces contain
the edges of the second.
[Here S<-a/3y = 0, S</?ya=0, S</a/?=0, and S^ay = 0, etc.
See the next Article and the Appendix to new edition of Elements of
Quaternions, vol. ii., note v.]
ART. 70. The square root of a linear vector function may be defined as
a linear vector function, which, operating twice in succession on any vector,
produces the same effect as the given function. " Writing then < for the
square root of the function <, we have, if y 1? y 2 and y 3 are the axes of
< , and if A 1? h z and 7* 3 are its roots,
Vyi-tfri. ........................ to
and consequently the axes of <* are also axes of < (see Ex. 2, Art. 66), and
the squares of the latent roots of <$r are the roots of <j>. In general, then, a
function has eight square roots answering to the double signs attributable to
#i^> #2 j ,9V* ^t does not follow conversely, that the axes of < are axes of <*.
As an example, let < have equal roots, and let it have indeterminate axes,
so that (<-<7i)(#7i"K ? /'}'2) := where x and y are arbitrary, g-^g^ being the
repeated root. A square root of the function may have three distinct
roots +g 1 , g\ , g%*. In this case there is an infinite number of square
roots, because we may select any vector #71+^72 to be an axis of <fF
corresponding to -f g^ , and any other vector yfy l -f-/y 2 may be selected as the
axis corresponding to -#i . For real square roots, the three roots g^ g 2
and # 3 must of course be positive.
The following resolution of a linear function <j> and its conjugate is
sometimes useful for example, in the theory of strain. It is due to Tait,
to whom is also due the conception of the square root of a linear vector
function.
Let i, j, Jc be the mutually rectangular axes of the self-conjugate function
<$', and let a 2 , 6 2 , c 2 be its roots. .Reducing <j> to the trinomial form
(Art. 62), ^aiSi'p+bjSfp + ckSttp, ............................ (11.)
where i\ /, K are to be determined, we have <f>'i=-ai', <V=-&/ and
<'=-<?'. These give <<^'= - o%\ t* - 6/ . S*y - ocife . St v #, but * is by
hypothesis an axis of <<', so that 6<f> f ia 2i i. Consequently we must have
i'*= -1, S'iy=S=0, and in fact t',/, k f form a mutually rectangular unit
system of vectors. Thus in particular ^i'=-ai, and <$><$' ~-a<$>i=+aH r y
and thus it follows that i' 9 / and Tc f are the axes of the new self -conjugate
function <'<, and that a 2 , 6 2 , c 2 are also its roots.
100 LINEAR VECTOR FUNCTION. [CHAP. vm.
Let q be a conical rotation which renders* ',/, If parallel to i,j\ &. We
have by (n.),
and therefore by the definition of a square root,
<}>P=(Wi*.q- l f><l and ^-^.(
and from these we also deduce
qpfL-VW^P ................................ (iv.)
In like manner we may prove that
and thus we can reduce the effect of a function < to a rotation preceded or
followed by the operation of a self-conjugate function.
ART. 71. We add one or two miscellaneous propositions respecting two
or more functions.
The functions <<, and (f>$ formed by taking the products of two functions
have the same symbolic cubic. For
and thus the functions have the same roots and the axes (y) of <,</> are
deducible from those of <<, by operating with <,.
In^ particular <f>,~ l <i><l> t has the same symbolic cubic as </>, and thus any
peculiarity in the nature of one function occurs also in that of the other.
Any two functions may be reduced simultaneously and generally in one
way to the forms
/) ......... (n.)
Assuming the possibility of the reduction, it appears that
<JVpv==a<l>V'iJw=iaaSX.fjLv, etc.,
and^thus the vectors VA/j, etc., are the axes of the function <t>" l <j> f and a b c
are its roots. If both functions are self -con jugate, we must have * '
b-c c-a a-b
and therefore for self -conjugate functions
, ......... (in.)
and further it is evident that
0, etc.
f the function
Va^ ...... (iv.)
V-
The invariant it is
be seoured fe y attributing
able "
ART. 7i.] A PAIE OF FUNCTIONS. 101
where m and m, are the third invariants of <j> and fa and where I and l t are
the two new invariants
ZSaj3y=2S<HW,7, Zay=2S<a<fc/:ty,y ................ (vu.)
Ex. 1. The locus of axes of the functions <l>+tfa where t is a scalar
parameter is the cubic cone
[If p is an axis <f>p+tfap=zgp. The surface represents a cone, as it is inde-
pendent of T/>.]
Ex. 2. The axes of functions of the family <j>+tfa form co-residual triads
on the cubic cone.
[The quadric cone SAp<p=0 in which X is arbitrary cuts the cubic in the
three axes of < and again in three lines in which it cuts SAp^p0, as we see
by substituting <p=#p+yA in the equation of the cubic. The remaining
intersection of the quadric cones is p || A. The cone SAp(<-K<,)p=0 passes
through the axes of $+tfa and through the three lines above mentioned, so
that these three lines are the residuals of every triad of axes (Salmon's Higher
Plane Curves, Art. 154). For other properties see Quaternion Invariants of
Linear Vector Functions, Proc. E.I.A., 1896.]
Ex. 3. Prove that the invariants I and l t are merely multiplied by a
scalar when <j> and fa are replaced by fa4>fa and fafafa-
[The scalar is the product of the third invariants of fa and fa. This very
general invariantal property leads to many theorems. See Phil. Trans.,
vol. 201, Part VIII., sections iii. and x.]
Ex. 4. Prove that the function $Vap='V(l>'- 1 a.fa'- l /3+Vfa'- l CL<l>'- l P is
co-variant with <j> and fa.
[Making the substitution of the last example, <'~ 1 becomes fa~ l <t>'~ l fa'~ l
and the function <3> changes into m 1 ~ 1 m%- l fa$fa.']
Ex. 5. If cr || V&p^ap show that p || Vfao-fa'o- ;
and more generally if cr is connected with p by the chain of relations
Pi II Vfapfap, p 2 || Vfapi<l>iPi, ... Or || Vfan^pn-^snpn-l,
prove that an analogous chain of relations connects p with cr.
[The second part of this example is related to the theory of the
Cremona transformations connecting vectors p and cr, the direction of a
vector (p) being connected by a one-to-one relation with that of a vector (cr).]
Ex. 6. If <Kp, is a linear and vector function of p and also a function
of the scalar t, the equation
represents a cone whose order is the number of values of t which satisfy
A being any constant vector.
Ex. 7. The equation
represents a surface which meets an arbitrary right line V(p-/3)y=0 in as
many points as there are values of t which satisfy
S(/3- a
LINEAR VECTOR FUNCTION. [CHAP. vm.
EXAMPLES TO CHAPTER VIII.
Ex. 1. Find the auxiliary functions x and ^ an d the invariants of the
function
Ex. 2. Invert the function <f>p+ VaVpa where <j> is a given function and
where a is a given vector.
Ex. 3. If < a p=cr 1 Va</> show that the conjugate function is
and prove that the spin- vector is e-^Vcr^'o.
(a) Show that the auxiliary ^ function of < a +c is expressible in either
of the forms ^ a Ba' l p + c(xp - V^'aVcr 1 /)) + c 2 p
or
and show that the third invariant of the same function is
(b) Prove that the axes of <f> a are determined by substituting a root of
the equation
Ex. 4. If ^p^p+aS/Sp, show that >/r,p==>/rp+V/:?<'Va/> and that
Ex. 5. Show that the ^ function and the third invariant of <p V/3Vap
may be reduced to the forms
and m -
Ex. 6. If <<.=<+ c, etc., show that
Ex. 7. Prove that
(a) Show that the conjugate of this linear function of p is V . <'V/3p . a,
and prove that the spin- vector is J^Va/S-aSe/? where is the spin -vector
of <.
(b) Show that the auxiliary ^ function is
(c) If V.<Vap./?=(7, show that p^^a-^'crCSa^)" 1 where x is an
arbitrary scalar. Deduce this result by the aid of the implied relations
Sap<jf>V=0, S/3<r=0.
Ex. 8. Prove that
V . <(p) = - 2V/3
where a, ^ and y are arbitrary vectors.
(a) Show that
V. TKp)= -2
(6) Express these quaternions in terms of the scalar invariants and the
spin-vectors.
ART. 7i.] EXAMPLES. 103
Ex. 9. Three lines are defined by the pairs of vectors (o^, rA (or^ r 3 ),
(0-3, r 3 ) as in Art. 36, Ex. 4, show that any line which is met by all tne
transversals of the given lines may be represented by
o-=<f>r where ST<J!>T=O,
the linear function < being defined by the equations
~1 = ^ T l <7 2 < T 2> 3 = $^3-
(a) The transversals of the same set of lines may be represented by
cr'=-<Y where Sr'<Y=0,
the function <' being the conjugate of <.
(&) "Writing
and expressing that the function <jf>( )- V/o( ) has a zero root, the locus of
the lines is found to be
S(/o - ><(p - )= ?n-hS<e
where m is the third invariant of the function <f> and where e is its spin-
vector,
(c) The same equation is satisfied by the transversals.
(d) Show that four given lines have in general two common transversals ;
and that these are determined by
o-'=-<Y where ST'(<r 4 -<T 4 )=0, Sr<Y=0,
the fourth line being defined by (o- 4 , r 4 ).
Ex. 10. Given any four pairs of vectors, (/?, CI H ), where w = l, 2, 3 or 4,
show how to find a linear vector function $ and a vector y so that <
Ex. 11. Given any six triads of vectors (y w , j8, ) where ?i=l, 2, ... 6 ;
determine two linear functions <^ and < 2 so that
Ex. 12. Verify by assuming p=a?a+yj8, SAa=0, 8X^=0, that the
solutions of the equations SA.p = 0, Bf><t>p = may be written in the form
where a is any vector perpendicular to A.
Ex. 13. Given two tetrahedra A'B'C'D' and ABCD, find a point E and a
function ^> so that
EA'^.EA, EB'=^>.BB, EC'=^>.EC, ED'=<J[>.ED.
(a) Show that corresponding faces of the tetrahedron determine with the
point E tetrahedra having a common ratio of volumes.
(5) If the lines joining corresponding vertices are generators of the same
system of a hyperboloid, it is possible to find four scalars , m, n> p so that
104 LINEAJR, .VECTOE FUNCTION. [CHAP. vin.
(c) These sealars are independent of the origin, and if the origin is taken
at the point B, we shall have
fa+mfl+ny+pS=0, la f +m/3'+ny+p8'*=0
for an arbitrary pair of tetrahedra, while if the lines joining the vertices are
generators of the same system of a hyperboloid, we shall have in addition
Ex. 14. Identify the expressions
fa
where t is a scalar variable, and show how to express the function <, and
the vectors A. and //, in terms of the vectors a, p, j and 8, and the sealars
, b, c and <f.
Ex. 15. Of what nature are the curve loci
and p =
Ex. 16. Gauss has described, in an unpublished MS. of the year 1819, an
operator which alters the size of any figure in a given ratio, and which turns
the figure through a given angle round a given line through the origin.
He proves that an operator of this kind depends on four numbers, that
successive operators compound into a single operator of the same kind, and
that the order of the operations is not commutative.
(a) Show that Gauss's operator may be expressed in quaternions by
cq( )g-\ c being a given scalar, and q a given quaternion.
(6) Hence prove his theorems.
(c) Compare and contrast the lack of commutation in the order of these
operators, or in the order of the operators 2 and cos. in the simple inequality
cos 2,# ^ 2 cos #,
with the lack of commutation in the multiplication of quaternions.
(d) Prove that the sum of two Gaussian operators is an operator of a
distinct kind.
(e) Prove that a sum of at least three Gaussian operators is required to
adequately express a linear vector function. (Bishop Law's Premium, 1 899.)
Ex, 17. Unit vectors a, B and y are directed respectively from the centre
of a regular solid to the middle point of a face (or to a vertex) ; to the middle
point of an edge of the face (or of an edge through the vertex) ; and to a
vertex on that edge (or to the middle point of a face containing the edge),
prove that 4 ^
y*=/2a,
where 7i=3 for the tetrahedron, w=4 for the cube and octahedron, and w=5
for the dodecahedron and ikosahedron.
(a) Hence show that all rotations which leave unchanged the region
occupied by the solid may be represented by powers and products of linear
vector functions A, K and i whicn obey the laws *
A,"=l, K 3 =l, i 2 =l, A,* tic, (rc=3, 4 or 5).
*See Hamilton on the Icosian Calculus, Phil. Mag., Dec. 1856; Proc. R.I. A.,
Vol. VI., pp. 415, 416. See also Burnside's Theory of Groups, Arts. 200, et seq.
ART. n.] EXAMPLES. 105
Ex. 18. A real linear function which is a symbolical n^ root of unity, or
which satisfies the equation
<fr n =l
is of the form /
$p . Sa/3y = (a cos ~ - /3 sin ~ )
where a, /? and y are arbitrary real vectors.
Ex. 19. The result of eliminating the vector T3 between the equations
may, when </> is self -con jugate, be expressed in the form
Sa^aS/o<^>p - mSap 2 = 0.
(a) In the same case,
S</oa
(6) And moreover *
* These examples are quaternion equivalents of the transformations in Arts.
383, 385 and 390 of Salmon's ^/ier P/owie CWes.
CHAPTER IX.
QUADBIC SURFACES.
ART. 72. If /(/>, p) is a homogeneous, rational and integral
scalar function of the second order in a variable vector p, so that
/3), ...(i)
where a and j8 are arbitrary vectors, the equation
/(/>, />) = const ............................ (II.)
represents a surface of the second order, referred to its centre as
origin. For by (i.) we find a quadratic in t which determines
two points in which an arbitrary line p = a+t/3 cuts the surface;
and on putting a = 0, the roots of the quadratic are equal and
opposite, showing that every chord through the origin is bisected
at that point.
The coefficient of t in (i.) is linear and homogeneous both in a
and in j8, and as it involves these vectors symmetrically we may
write
(III.)
where <p is a self-conjugate linear vector function. Thus the
equation of the central quadric is expressible in the form
f(p, p )-Sp<f>p = const ......................... (iv.)
Without loss of generality we may suppose the constant incor-
porated in 0, and we take as the equation
Spft>=-l, .............................. (v.)
in which, as we have said, </> is self-conjugate. Of course, and
without gain of generality, we may suppose <f> not to be self-
conjugate in (v.), for the spin-vector automatically disappears
from an equation of this form (Art. 67) ; but this i,s very likely
to lead to mistakes in further developments, and it adds needless
complexity.
ART. 73.] QUADRIC SURFACES. 107
ART. 73. Equation (v.) of the last article gives
=-l or -SU P 0Up--=l ......... (i.)
if r is the length of the central radius parallel to Up.
For a closed quadric, an ellipsoid or sphere, r 2 is always
positive, as every line through the centre meets the closed
surface in real points. For a hyperboloid, the radius becomes
infinite for an edge of the cone
SUp0Up=0 or Sp0p = 0, .................... (II.)
the asymptotic cone of the surface. The sign of the expression
r -2 or _ SUp^Up changes on passing through a zero value, and
the expression remains with changed sign until it passes again
through a zero value. So on one side of the cone Sp<p = 0, lines
meet the hyperboloid in real points, and on the other side the
points are imaginary and the corresponding vectors are of the
form p = x/ :r V, (Up=Up', Tp = >v/^lTpO, where p' is a real
vector.
The vectors p' terminate on the quadric
Sp0p= +1 ............................. (in.)
the conjugate of the quadric Sp<p= 1.
For the sake of brevity we shall write generally r 2 for the
square of the length of the radius whether that square be
positive or negative, the interpretation in the latter case being
that just given.
An arbitrary right line p = a+/3 cuts the quadric Sp<p= 1
in the points determined by the roots t of the quadratic
-1 .................. (IV.)
For a real and positive root, the point is in the direction +Uj6
from the extremity of a, and for a negative root it is in the
direction U/3. For equal roots, the line touches the surface;
and for imaginary it cuts it in imaginary points.
The locus of the middle points of chords parallel to /3 is the
diametral plane
................................ (v.)
for if a is the vector to any point in this plane, the roots of (iv.)
are equal and opposite. If the diametral plane of /3 contains the
vector a, that of a contains /3 in virtue of the self-conjugate
property of <p, for then
=0 ........................... (vi.)
The equation has equal roots if
0, ............... (VII.)
10 s QUADKIC SURFACES. [CHAP. ix.
and regarding /3 as variable, this is the equation of the tangent
cone Irom the extremity of the vector a referred to that
extremity as origin, for it is independent of T/3. Replacing
/3 by />-a, the equation of the same cone referred to the centre
as origin easily reduces to
0', ......... (vm.)
and the form of the equation shows that the cone touches the
quadric along its intersection with the plane
S/o0a= -1 .............................. (IX.)
the polar plane of the extremity of a.
If the vector a terminates on the surface, the equation of the
cone becomes the square of the equation of a plane the tangent
plane at the extremity of a,
Syo0a=-l, Sa0a=-l ....................... (x.)
Allowing on the other hand a to vary arbitrarily in the quad-
ratic equation, and putting for greater clearness a p' = p tf/3,
the vector p being drawn from the extremity of the vector tfj
while p is drawn from the centre, we see that
S P '<j>p'=-l-t*S/3<p/3 if 8/0/3=0 .............. (XL)
These two equations jointly represent the section of the quadric
by the plane S^/S^S/fy/?, ........................... (xn.)
and the centre of the section is the origin of vectors //, or the
extremity of the vector t/3. Hence the locus of centres of
sections by planes parallel to (v.) is the line through the centre
parallel to /3, as indeed might have been proved directly from (V.).
The section (xi.) is similar to the parallel central section of the
quadric, for if r' is the radius of the section parallel to p and r
that of the quadric,
= l-^p^ (xin.)
if V is the radius of the quadric parallel to /3.
The equation of the normal to the quadric at the extremity of
the vector a is
p = a+x<pa, or V(/> a)<pa = ; (XIV.)
and the normals which pass through a given point /? are six in
number and are determined by the equation
/3=p+x<f>p, or V(/3/o) 0/o = 0, and Sp<f>p= 1. ...(xv.)
To solve these equations we have
/>=(l+cc0)~ 1 j 8, where S/30(l+&0)~ 2 /3 = 1, (xvi.)
because S/>0/>= 1, and on inversion we find 'a sextic equation
n cc.
ART. 74.] TANGENTS AND NORMALS. 109
Ex. 1. Prove that the rectangle under the intercepts from the extremity
of a on the line p = a+ tft is
where a! and b' are the central radii parallel to a and ft.
Ex. 2. The ratio of the rectangles under the intercepts of lines drawn
from a fixed point is independent of the position of the point, and is equal to
the ratio of the squares of parallel central radii.
Ex. 3. Chords drawn through a point are divided harmonically by the
quadric and the polar plane of the point.
[Put =4 + 4 where * x and t z are the roots of the quadratic (iv.).]
Ex. 4. Find the central vector perpendicular on the tangent plane at
any point, and obtain the locus of the feet of central perpendiculars, or the
central pedal surface.
Ex. 5. Prove that the central pedal surface is the inverse of the reciprocal
quadric.
Ex. 6. Prove that the ratio of the perpendiculars from a point A and
from the centre on the polar plane of B is equal to the ratio of the perpen-
diculars from B, and from the centre on the polar plane of A.
Ex. 7. Find the locus of the poles of tangent planes to the surface
= - 1 with respect to the surface SjO^>2/>= - 1.
Ex. 8. Find the pedal surface for an arbitrary point.
Ex. 9. The feet of the normals which pass through a given point are the
intersections of a twisted cubic with the quadric.
[Compare (xv.) and Art. 65, Ex. 8, p. 93.]
Ex. 10. The normals through a given point lie on a quadric cone
S(p ft)<j>j3ct>p=Q, and the feet of the normals lie on the cone Sp/></)=0.
(a) Both these cones have edges parallel to the three axes.
Ex. 11. Find the condition of the intersection of normals at two points
a and /J.
Ex. 12. Find the equation of the polar plane of a to the quadric
Sp<t> 1 4> 2 p= - 1, <x< 2 being the product of two linear functions.
[Note that fyfa' is the conjugate of <i< 2 -]
Ex. 13. Prove that the polar line of pa+t/3 with respect to the
quadric Sp</>p= 1 is
ART. 74. The central plane SXp = is the diametral plane of
chords parallel to $~ l \, as appears on comparison with (v.) of the
last article. The locus of the centres of sections by planes
parallel to SX/> = is the right line
X = .................... . .......... (I.)
HO QUADEIC SURFACES. [CHAP. ix.
The vector to the pole of the plane (Art. 73 (ix.))
SX/9=-l is 0- a X; ........................ (ii.)
and the plane touches the quadric if (Art. 73 (X.))
SX^X^ -1, .......................... (in.)
and as X varies this is the tangential equation of the quadric.
But SX/>= 1 is the polar plane of the extremity of X with
respect to the unit sphere, Tyo = l or /> 2 = 1, and the equation
(in.) may therefore be regarded as that of the reciprocal of the
quadric with respect to the unit sphere.
The vector to the centre of the section by SX/> = - 1 is by (i.)
the tensor being determined so that this vector may terminate in
the plane S\p = 1 ; and on comparison with (xin.) of the last
article, the ratio of the radii is given by
.
^ }
Ex. 1. By direct comparison of SAp-f 1=0 with (xn.) of the last article,
find the vector (iv.) of the present.
Ex. 2. Find the reciprocal of the surface with respect to an arbitrary
sphere.
Ex. 3. Find the lines in which the plane SXp=0 cuts the cone Sp<p=0 ;
and show that they are parallel to
where a is an arbitrary vector in the plane.
[Assume the lines to be a+ta! where Vaa' = A. and actually solve for t on
substitution in the equation of the cone.]
Ex. 4. Prove that the tangent of the angle between the lines in which
the plane SAp=0 cuts the cone $p<f>p=ti is
[If a -Kj a' and a+ 2 a' are the lines, calculate a 2 + (jf 1 -h^)Saa / + y < >a' 2 and
Ex. 5. Show that the lines in which the plane SAp=0 cuts the cone
Sp<p=0 are parallel to the vectors
<
AKT. 75. The vector radii a and ft of the quadric are con-
Fg ateif Sa0/3 = 0, ................................ (i.)
that is if one lies in the diametral plane of the other (Art. 73
(vi.)) ; and it follows geometrically, or directly from the equations
of the tangent planes
-1, ..... (IL)
ART. 75.] PRINCIPAL AXES OF SECTION. Ill
at the extremities of these vectors, that each vector is parallel to
the tangent plane corresponding to the other.
If the vectors are perpendicular as well as conjugate, they are
the principal axes of the section by their plane, and the condi-
tionsare Satf>/3 = Sa/3 = ......................... (ill.)
From these we see that
...................... (IV.)
so that if one vector is given, the other is determinate ; or given
that a line is to be the principal axis of a section, the other prin-
cipal axis is determined by (iv.), and the normal to the section is
parallel to
Va/3 |! aVa^a || <f>a . a 2 -aSa0a || 0aTa 2 -a ........... (V.)
Thus to determine the principal axes in a central plane SX/> = >
we have
0aTa 2 -a||X or a || (^Ta 2 -!)-^; ............. (VI.)
and because SXa = 0, we have if Ta 2 = r 2 ,
l)- 1 X = or ^SX^X-r 2 SX x X+X 2 =0 ..... (vn.)
using the formula of inversion (Art. 65). Thus a quadratic in r 2
is obtained and substitution of its roots in (<r 2 1) -1 X gives the
directions of the vectors required.
The principal axes of a surface are normal to the tangent
planes at their extremities, so that
V/%) = ............................ (vm.)
for a principal axis. These are the axes y x , y 2 , y s of the
function <p.
Ex. 1. Find the maximum and minimum radii in a central section.
[Here SA.p=0, Sp<p= 1, Tp=max., and on differentiation, SXd/)=0,
S<pdp=0, Spdp=0, so that the three vectors A, <j>p and p are coplanar, or
Operating by Sp, we fall back on (vi.).
Ex. 2. Find the maximum and minimum radii of the quadric, and show
that their directions are the solutions of
Ex. 3. The sections by planes perpendicular to A, are rectangular
hyperbolas if
Ex. 4. The equations (iv.) fail in one case.
[Where the vector a is a principal axis of the surface.]
Ex. 5. In general, the three radii are coplanar which are axes of sections
having any three mutually rectangular radii as the remaining axes.
[Because <f> is self -conjugate, Va- 1 <#>a+V/3- 1 <^/?+Vy- 1 ^y=0 if a, (3 and y
are mutually perpendicular (Art. 6*7, Ex. 8, p. 97).]
112 QUADRIC SUEFACES. [CHAP. ix.
Ex. 6. The sum of the squares of the reciprocals of three mutually
rectangular radii is constant.
Ex. 7. Interpret geometrically the equation
which asserts that the plane SAp=0 cuts the quadric in a section having a
principal axis equal to r.
[This expresses that the plane touches a certain cone.]
Ex. 8. Central planes cut a quadric in sections of given area A . Prove
that their envelope is the cone
Ex. 9. The axes of the section by the plane SAp+l=0 are the roots of
the quadratic
Ex. 10. The area of the section made by the plane SA/>+ 1 =0 is
ART. 76. From any pair of conjugate radii a and ft we can
derive a third radius conjugate to both so that
...................... (I.)
We may in fact regard the two conditions in y as equations of
planes, and
With proper tensor the radius y is
t-tVaft , N
7 -SVa^V ................... (IIL)
In terms of the three mutually conjugate radii, the equation
of the quadric is
(iv.)
as appears on substituting yo=EaS/3y/) : Sa/3y in S/>d>p= - 1 and
attending to the conditions.
Writing (compare Art. 70)
a = 0-V, ft = <f*p, y = f*y' ................. (y.)
it appears by (i.) that the vectors a', /3 7 and y' are mutually
perpendicular, and because a, ft and y terminate upon the
surface S/>0p=-l, it further appears that a, ft" and y are
unit vectors. The theorems of Art. 69 therefore apply, the
vectors a, ft and y being the results of operating by a linear
ART. rr.] CONJUGATE EADIL 113
function (0~*) on three mutually rectangular unit vectors.
Thus the sum of the squares of three mutually conjugate radii
is constant, etc.
Ex. 1. The radii a, /!?, y being mutually conjugate, prove that
^ a=
i j-i j-
ana that
Ex. 2. The locus of the extremity of the diagonal of a parallelepiped
having three mutually conjugate radii as conterminous sides is
Ex. 3. The locus of the mean point of a triangle formed by the
extremities of mutually conjugate radii is
Ex. 4. The locus of a point from which it is possible to draw three
tangents parallel to mutually conjugate radii is
8^+1=0.
Ex. 5. In the last example show that a point on the locus is
and that the points of contact are the extremities of
ART. 77. To find the cyclic planes of a quadric we have to
throw its equation into the form
-1 ..................... (i.)
or to determine g, \ and /* so that for all vectors p,
(ii.)
It follows that (pg must reduce every vector to a fixed plane,
that of X and p. The scalar g must therefore be one of the
latent roots of ^>, say g =g% and in terms of the axes,
(m.)
because <f>p = - igfiip jgfijp -
Thus
t\
J.Q. H
114 QUADEIC SURFACES. [CHAP. ix.
where t is arbitrary. The transformation is real only if
The cyclic planes SAp = 0, 8^/3 = cut the surface in circles of
radius g^, and these circles are real only if # 2 >0.
The planes SAp-fZ = 0, S/z/>+m=:0 cut the surface in circles
lying on the sphere
Sp<f>p + 1-2 (SXp + Z) (Sup + m) = 0,
or gp* - 2Sp (Ip + mX) - 2Zm + 1 = 0.
In nearly every problem relating to quadrics some valuable
information will be gained by throwing the equation into the
cyclic form or into the focal form of the next article. This
transformation is not generally of any great difficulty.
Ex. 1. Reduce a quadric to the form
[This gives Dr. Salmon's focal property. The locus of the extremity of
the vector a is a hyperbola the focal hyperbola, and this depends on
equation (iv.).]
Ex. 2. Prove that the roots for Hamilton's cyclic form are
g,
Ex. 3. Any two circular sections of opposite systems lie on the same
sphere.
Ex. 4. If a quadric is a surface of revolution,
for all vectors p.
[The self-conjugate function <f> has two equal roots (c) and (Art. 66
p. 95)
is identically zero for all vectors a and /5, or ^p-
Ex. 5. If for all vectors p
the quadric is of revolution.
Ex. 6. From a fixed point A, on the surface of a given sphere, draw any
chord AD ; let D be the second point of intersection of the same spheric
surface with the secant BD drawn from a fixed external point B - and take
a radius vector AE, equal in length to the line BD', and in direction either
coincident with, or opposite to, the chord AD : the locus of the point E
will be an ellipsoid, with A for its centre, and with B for a point of its
surface. r
[Elements of Quaternions, Art. 217 (6). See also Lectures, Art. 465. If c
is the centre of the sphere, the isosceles triangle AOD gives =K , or
CD = - AD- 1 . c A . AD = - AE' 1 . CA . AB, and therefore
ART. 78.] CYCLIC AND FOCAL EQUATIONS. 115
,
reduces immediately to Hamilton's cyclic form.]
Ex. 7. Conceive two equal spheres to slide within two cylinders of
revolution, whose axes intersect each other, in such a manner that the right
line joining the centres of the spheres shall be parallel to a fixed right line ;
then the locus of the varying circle in which the two spheres intersect each
other will be an ellipsoid, inscribed at once in both the cylinders.
[Hamilton, Lectures, Art. 496. Taking the spheres to be T(p-fo)=5,
T(p- /?)= 6, where a, ft and b are given and where t is a variable scalar,
we find on elimination of ,
ART. 78. To find the right circular tangent cylinders of a
quadric, observe that if the vertex of the tangent cone (Art 73
(vni.)) passes off to infinity, the equation of the tangent cylinder
parallel to a is
2 = ................... (I.)
A right circular cylinder parallel to a and of radius TcT 1 is
represented by
TVap = l, or (Vap) 2 +l = 0, .................. (ll.)
and identifying this with (i.) we have to satisfy
s ^=%)? + w .................... < m >
for all vectors p, or what is equivalent we must identify
<t>CL&pd>Q, Tr /,, \
^ = %^T- aVap ...................... (lv ->
This is identical for p = a ; and for p = <j>a we have
(v.)
Here then is a linear relation connecting the vectors 2 a, 0a
and a, and it follows (Art. 66) that a must be coplanar with a
pair of axes, i and k suppose, and that (say)
This gives on comparison with (v.)
Sa0a=-gr 8 ft, Sa^a = (g 3 +g l + a z )8a<j>a } Saj = 0, ...(VI.)
and putting p=*j in the identity (TV.), we find
a 2 =-& ............................... (vn.)
The identity is now satisfied for three non-coplanar vectors,
116 QUADBIC SURFACES. [CHAP. ix.
j, a and <j>a and therefore for all vectors; and if 0/3 = U0a, the
equation of the quadric is by (in.) reduced to
Sp0p = 6(SpU/3) 2 +a(VpUa) 2 = -1, ............ (VIIL)
where a=# 2 , &=flr 2 --0i-08
which is Hamilton's focal form, if we remark that by (vi.)
and (vn.)
If a=ix+ltz we have by (vi.) and (vn.)
01-08
ART. 79. To jfeci the generators of a quadric, we express
that when we substitute p+ta in its equation, the equation is
satisfied for all values of t. Thus
................ (i.)
From the second and third of these
Vap=#0a, or p^xa'^a + ya, ................ (ll.)
and substituting for p in the equation of the quadric,
or simply 2 m = - 1. Thus the equation of the generator is
T
.................... (ill.)
it being implied by the form of this equation that Sa~V = -
Generators of one system correspond to the sign +, and those
of the other system to the sign .
Ex. 1. Prove that generators of opposite systems intersect.
Ex. 2. Find the locus of the feet of central perpendiculars on the
generators.
[From the equation p= /y _I a -i^ a we find a || Yp<p, and substitution
in Sa<a=0 gives a quartic cone which intersects the quadric along the
locus.] &
- Ex- 3. Prove that the locus of intersections of generators which cut at
right angles is the intersection of a sphere with the quadric.
[Note that a central plane parallel to a tangent plane cuts the asymptotic
cone in lines parallel to the generators.]
ART. so.] GENERATORS AND CENTRES. 117
Ex. 4. The locus of intersections of generators which cut at a given
angle is
[See Ex. 4, Art. 74.]
ART. 80. When the equation of a quadric is given in the form
Sp0/>-2S/>+Z=0, .......................... (I.)
in order to find its centre, or centres, we may replace the
equation by
Z = 0, ...(ll.)
and if co terminates at a centre the part linear in p co vanishes,
and co is a solution of the equation
0o> = e. ................................ (ill.)
Operating by \js we have
mco = -y/re, ............................... (iv.)
and the vector to the centre is finite and determinate if m is not
zero. If m is zero and -^e not zero, the centre is at infinity in
the direction of ^e, and the surface is a paraboloid. If \fse is
zero, m must also vanish, and the solution is (Art. 65)
w'fo = xe+'*/'a>, T/re = 0, ....................... (v.)
and the surface has a line locus of centres and is a cylinder,
v/rco being parallel to the axis of $ corresponding to its zero root,
and the length of T//-CO being indeterminate. If <ni' vanishes,
the function \fs vanishes identically since <p is self-conjugate
(Art. 67), and in fact <f> is of the form ai&ip. If %e is not
zero, the line of centres is at infinity since (v.) can only
be satisfied for infinite values of CD. If however x == 0> the
solution is wi ff = e+x, X = 0, ....................... (vi.)
and the surface is a pair of parallel planes. More simply when
^ = aiSia) = e and ^ ae+ aiSie = 0,
equation (in.) becomes aSico = Sie.
In the case of the paraboloid, equation (v.) without the
condition i/re = 0, or
\^ e II & 0& = ............... (vil.)
is the equation of the axis, remembering that \fsw || 7c=u/c where
U is an indeterminate scalar. We have in fact on operating
by 0, m'($ e)= ^e, and the term linear in p co is propor-
tional to S/c(p o>). In like manner it may be shown that (vi.)
without the condition x e = ^ represents the axial plane of a
parabolic cylinder.
118 QUADKEC SURFACES. [CHAP. ix.
ART. 81. We propose in this article to give a short account
of the cone of the second degree and of sphero-conics. (See
Elements of Quaternions, Art. 1 96.)
The equations
represent respectively a plane and a sphere which passes
through the origin of vectors. Combining these equations so as
to eliminate T/>, the equation
+l = 0, or SapS@p+p* = 0, or S/3 / oS + l = J ...(IL)
represents the cone whose vertex is the origin and which passes
through the circle of intersection of the plane and sphere.
The third form of the equation shows that the cone passes
through a second circle, the circle common to the plane and
sphere Sfr+i-o, ss-i-o, ..................... (m.)
and thus exhibits the theorem of Apollonius that an oblique
cone having a circular base has a second series of circular
sections.
The second form of the equation shows that the product of
the cosines of the angles between an edge of the cone and the
cyclic normals (Ua and U/3) is constant, for this is
SU.a/>SU./3p = Ta" 1 J 8- 1 ; .................. (iv.)
or what is equivalent, if the cone is cut by a sphere concentric
with the vertex, the product of the sines of the arcual perpen-
diculars let fall from any point of the sphero-conic of intersection
on the two cyclic arcs (the great circles in the planes Sap = 0,
S/3p = 0) is constant.
If Up and TJ/>' are the vectors to any two points P and P' on
the sphero-conic, and if the great circle PP 7 cuts the cyclic arcs
in Q and Q', it follows from the second of equations (li.) that
U(TJ/>SaU0' Tlyo'SaU/o) is the vector to one of the points (Q)
and that U(U/oSaU/o TJp'SdUp) is the vector to the second
Sint (Q'), Q being in the cyclic plane Sa/> = and Q' in S/3/o = 0.
3nce, from the form of the expressions for the vectors to these
points, we learn that the arc PQ is equal to the arc P'Q'
If P 7 and P" are two fixed points on the sphero-conic, and if P
is a variable point likewise on the conic ; if the arcs PP' and PP"
cut one cyclic arc (Sap = 0) in Q' and Q", the length of the arc
Q'Q" is constant. This follows most easily by producing the
radii of the points P, P' and P" to meet the plane Sap + l =
of equation (i.) in the points P , P ' and P ". It is evident that OQ'
and OQ" are respectively parallel to P P ' and P P ", and more-
ART. si.] SPHEEO-CONICS. 119
over the angle P O 'PO P O" is constant since it is the angle subtended
at a point on the circumference of a circle by two fixed points
likewise on the circumference.
Given the cyclic arcs of a sphero-conic and a point on the
conic, the conic is determined by elimination of t from the
equations
SapSpp + tp* = Q, SayS/3y+y 2 = 0, T> = 1,
the vector y terminating at the given point, and for convenience
the radius of the containing sphere being taken equal to unity.
The three propositions just proved are used by Hamilton to
establish the associative principle of multiplication of quaternions.
In the figure the great circles GLIM, CHBG, DAEC are the traces
of the planes of three versors
OL __ OH OC
M I
~OG' '' OC' 5 ~OE*
Constructing the product r$ = OH:OE, the point H is deter-
mined and the sphero-conic HKBF is drawn through the point H
having GLIM and DAEC for cyclic planes. Producing the arcs
CH and EH, the points B, G, IT and I are constructed. The point
L is joined to B and LB is produced to K and A. The arc FK
is drawn and produced to M and D. It follows then that the
arcs GL and IM are equal and also the arcs CE and AD, and
moreover FM = DK and AK = BL by the properties of the sphero-
conic.
We have therefore
= 25= 2I-2M 2I=2^=2:!E=25: 2^
7- rs ~2- QE ~2'- OF - OI OF~OF~OD~OA' OD '
OK OL OL OG
By proving the properties of the sphero-conic without employ-
ing the associative principle, this principle is established since
we can show that for any three quaternions q.rs = qr.s.
In addition to the properties just proved for the sphero-conic, it is easy
to see that great circle arcs which intersect at a point on the curve include
supplemental arcs (such as CA and GL, Fig. 25) between the points in which
120 QUADKIC SUBFACES. [CHAP. ix.
they cut the cyclic arcs. Reciprocating these properties, the cyclic arcs
become the foci E and F (Fig. 26) of the reciprocal sphero-conic, and if the
two foci and one tangent arc AB are given, the conic can be constructed. If
from any point on the sphere, two tangent arcs are drawn to the curve and
also two focal arcs to the foci, then one focal arc makes with one tangent
the same angle as the other focal arc makes with the other tangent. ^ More-
over opposite arcs of a spherical quadrilateral, ABCD, circumscribing the
conic subtend supplemental angles at the foci.
FIG. 26*
From these properties Hamilton deduces the associative principle. The
versors q and r are represented by the directed angles BAE and EBA, and their
product qr is (Art. 30, Ex. 5, p. 30) represented by the external angle at E
or by the equal angle CED. A third versor s is represented by DOE, and the
external angle of the triangle DEC represents the product qr-s (namely,
qr into $). Making FCB and CBF respectively equal to the angles of s and
of 5', the point F is found ; and when the sphero-conic having E and F for
foci and AB for tangent is constructed, it follows that BC and CD are also
tangents on account of the equality of the angles marked r and of the angles
marked s. Again, because CED was constructed equal to the supplement of
ABB, the arc DA will be a tangent to the curve, and PAD will be equal to the
angle of q, and DFA will be supplementary to CFB. Hence FAD and DFA
represent respectively q and rs, and the external angle of the triangle ADF
represents the product q . rs. But the angle between DA and DF is equal to
the angle between DC and DF, and therefore q.rs=qr.s.
To find the locus of a point on the surface of a unit sphere,
the sum of whose arcuaL distances from two fixed points, E and
F, is constant, we have in the first place for the cosine of the
sum of the arcs,
SU.epSUj7 / )-TVU.e / )TVU. ?? p = cosa; (v.)
or on rationalization, we find the locus to be a sphero-conic,
(SU. / o) 2 +(SU.i 7/ o) 2 -2cosaSU.6 / )SU^==sin 2 a; (vi.)
since (SU.e/>) 2 +(TVU.e/>) 2 =l. (Compare Elements of Qua-
ternions, Art. 360.)
This may also be written in the form
(vii.)
ART. 8-2.] CONFOCALS. 121
so that the sine of the arc between a point and a focus is propor-
tional to the sine of the perpendicular on a directrix arc.
Many interesting examples and illustrations will be found in
the Elements, Book II., Chap. III., Sections 1 and 2, and in
Art. 306, and also in the sixth of the Lectures on Quaternions.
Ex. 1. Through three given points on the surface of a sphere, it is
required to draw a sphero-conic so that a given great circle shall be one of
its cyclic arcs.
[If y l5 y 2 and 73 are the vectors to the three given points, it is necessary
to find p so that S/3/>Sap+p 2 =0 may be satisfied on replacing p by y 19 y 2
and y 3 , a being a given vector. The vector ft is given by
Ex. 2. Find the relations between the cyclic normals of a cone and its
focal lines.
[Identifying (vi.) with the second form of (n.), the required relations are
easily obtained.]
Ex. 3. Prove that
S . V . VojSVSeV . V/tyVepV . YySV/oa^O
represents the cone which has five edges parallel to five given vectors,
a, /?, y, Sj e, and show that the form of the equation furnishes a proof
of Pascal's property of the hexagon inscribed to a conic. (Lectures on
Quaternions, Art. 442.)
CONFOCAL QUADRICS.
ART. 82. Quadrics of the family
S/>(0+aO/>=-l, ........................... (L)
in which x is a variable parameter, are called concyclic, as they
have common planes of circular section (Art. 7*7).
The reciprocal system of quadrics
is called a conf ocal system.
Because we may write (II.) in the form
Sp(\fs+x x +x?)p= -(m+m'aj+mV+a?), ......... (in.)
it appears that three quadrics (u.) pass through an arbitrary
point; and reciprocally, three quadrics (i.) touch an arbitrary
plane. Also one quadric (i.) passes through an arbitrary point,
and one quadric (II.) touches an arbitrary plane.
Conf ocal quadrics cut at right angles. Let cc, y and z be the
parameters of the three quadrics which pass through an arbitrary
point (a). Then
122 CONFOCAL QUADRICS. [CHAP. ix.
for all functions <j>+x, (</>+x)~ l , etc., have the same axes, and
are therefore commutative (Art. 66, Ex. 2, p. 95). Thus at any
point of the intersection of the quadrics x and y,
S P (<j>+x)- l (<}> + y)- l p = Ql ................... (iv.)
which expresses that the normals (Art. 73, p. 108) (<f>+x)~ l p
and (<f>+y)~ l p are at right angles.
Ex. 1. Beduce So . _!!L_ . p to a sum of the form
^
[We may employ the method of partial fractions, and treat <j> as a scalar.
it being commutative with scalars and with <-h#, etc.]
Ex. 2. If x, y and z are the parameters of the confocals through the
extremity of the vector p, the expressions
are respectively equal to
, zero, and -
x z x
Ex. 3. Prove that
are the principal axes of the central section of the quadric # made by the
plane parallel to the tangent plane at p.
Ex. 4. Find the centres of curvature at a point on the quadric #, and
prove that they are the poles of the tangent plane to # with respect to the
confocals y and z.
[If y is the vector to a centre of curvature, two consecutive normals
intersect at its extremity, or yp + t(<t>+M)~ l p is stationary when p and t
vary. Therefore
Operate with S^+^p, and Sp(^+^)- 1 (<#>+^+#)"- 1 p= j an d on
comparison with (iv.) the roots of this quadratic in t are seen to be y-$ and
s-x. Therefore y =(<+#)(< -M)- 1 ^ Y'=($+z)(4>+x)- l p are the vectors to
the two centres. Observe that dp is also tangential to the quadric z.
Compare Art. 87, Ex. 1, p. 136, for the method employed.]
Ex. 5. If X) y and s are the parameters of the three confocals through
the extremity of the vector p, prove that
Ex. 6. Prove that the plane SAp 4-1=0 touches a confocal at the
extremity of the vector
and show that the locus of points of contact for a system of parallel planes
is a rectangular hyperbola.
ART. S3.] NORMALS AT A POINT. 123
Ex. 7. Prove that the locus of points of contact of planes through a line
is a twisted cubic.
[Put for A in the last example (X + ^Xl + 0" 1 and verify that an
arbitrary plane meets the curve in three points.]
Ex. 8. The locus of the poles of a plane with respect to a system of
confocals is a right line.
Ex. 9. The locus of the poles of planes through a given line is a hyper-
bolic paraboloid.
=(<+ w) ( A -M/iXl + O" 1 is the locus of a line dividing two given lines
similarly.]
Ex.10. The plane S/>AcA=0
is the locus of poles of planes perpendicular to A.
ART. 83. In many investigations relating to the confocals through a
given point, the extremity of the vector a, it is convenient to employ the
vectors
A = (<#> + .r)-ia, fL-fa+^-ia, v = (<+*)-i a , .................. (i.)
which when originating at the centre terminate at the reciprocals of the
three tangent planes. These vectors are of course normal to the three
confocals. We have then
a=(<+tf)A = (<+y)//, = (<+;s),/, SAa = Sjua = Si/a=-l; ........ (ii.)
and because these equations give
it follows that
Sjuv=SvA = SV=0, ............................... (m.)
or confocals cut at right angles.
We also have from the same equations
^=f t +(y-^)(^+^)~ 1 ^ etc., ......................... (iv.)
so that //,H(y-^S//,(<H^~V=0, Cy-^SK^ + aO'V^O,
or (x-yY l = -STJ>(<+tf)-iTJ>= + SUA(< + ?/)- 1 UA, etc.,
S/^(<^4-.t 1 )~ 1 ^ = 0, etc .................................. (v.)
And the axes of the section of the quadric parallel to the tangent plane
are \lx-y . Uju, *Jx z . Uv ; and those of the section of the quadric y
parallel to its tangent plane are <jya; . UA, \ly-z. Ui/.
Introducing a new self-conjugate function defined by the equation
0/) = </> + aSap, .................................... (vi.)
we may replace equations (11.) by
(0+ai)*.=(0+y)iL=(0+z)v~Q t ........................ (vn.)
so that A, //, and v are the axes and #, y and z the roots of this function.
If So>p= 1 is the equation of any plane through the point a, and if 57
is the^pole of the plane with respect to any confocal u,
a>, or T-a = (0-M)to, .................... (vm.)
because a=+aS<xo>. If the plane touches the quadric u, the pole lies in
the plane, and the vector G7-a (joining two points in the plane) is normal
to co. Thus in order to determine the point of contact of the plane
124 CONFOCAL QUADBICS. [CHAP. ix.
Scop = -1 and the parameter of the touched quadric, it is only necessary
to operate on o> by the function 6 and to resolve #o> along and at right
angles to o> ; f or
Qto=toVto-iQu + u&v-iQto=r3-a.-uu', tJ=a4-coVa>~ 1 ^a), w= - Sor 1 0co. (ix.)
The vector aT being still supposed to terminate in the plane, the vector
5T-a(=T) is tangential to the surface u and perpendicular to o>. Hence as
57 varies subject to the condition SsTo) = Saa)= -1, we find by (vm.) that
S(&-a)(6.+u)- l (G~a) = 0, or Sr^+^r^O .............. (x.)
is the equation of the tangent cone from a to the confocal ^, referred in the
first case to the centre of the quadrics and in the second to the extremity
of a. The form of the equations shows that the tangent cones drawn from
a point are confocal. They intersect in pairs along any line through the
point, for (x.) may be replaced by
0, .............................. (xi.)
and may be regarded as a quadratic determining the quadrics touched by a
given line (Ur= const.); and they intersect at right angles by the general
property of confocals.
We can thus determine the two quadrics touched by an arbitrary line.
Ex. 1. Prove that
Ex. 2. A right line defined by the vectors cr and r of Art. 36, Ex. 4,
touches the confocals whose parameters are the roots of the equation,
Ex. 3. The lines through a given point touching confocals with a given
sum of parameters, generate the reciprocal of the tangent cone to a fixed
confocal.
[The cone of the lines is Sr^-m^-a 2 - y)r=0, if v is the sum of the
parameters.]
Ex. 4. If v and v are the vectors to the reciprocals of the tangent planes
of the confocals u and u' at the points A and B, and if r is the vector AB,
[Here T=(<J>+U')V' ~'(<$>+u)v. This is Gilbert's theorem.]
Ex. 5. If the points A and B are both points of contact of the line with
the quadrics,
Si/i/=0,
ART. 84. There is a third general method which is often useful for
dealing with the properties of confocals. Writing the equations of the three
confocals through a point in the forms
we are led to assume
as an expression for the vector to the point of intersection. The square roots
#)*, etc., are commutative, and, accordingly, on substitution in
(m.)
we find
ART. 84.] ELLIPTIC COORDINATES. 125
This is identically satisfied, for the confocal x as well as for the confocals
y and 2, if
2 =0, Se<#>=0, S < 2 e=-l; .......................... (iv.)
or what is equivalent, if
2 =0, Sx=0, Se^=-l; ........................... (v.)
that is, if e is the vector to a point of intersection of three known surfaces,
one of which is of course imaginary. Therefore (n.) coupled with the con-
ditions (iv.) or (v.) ^is the vector to a point of intersection of the three
confocals ; and allowing any two of the parameters, y and z, in (n.) to vary,
the vector equation represents the surface x ; if only one parameter (#)
varies, the equation represents the curve of intersection of the confocals
y and a.
Again, we may differentiate p, regarded as a function of #, y and z, as
given by equation (n.) just as if < were a scalar, and we have
and the method easily lends itself to the treatment of lines traced on a
quadric surface.
Ex. 1. Prove that the vectors (< + x)~ l p, (<f> +y)~ l Pi (<l> + z)~~ l p are mutually
rectangular, and that the squares of their tensors are
where m(^)=m+m'.r+mV+^ 3 , and where x> y and z are the parameters of
the confocals through the extremity of p.
[Using (u.), we have Sp(^+y)- 1 (^>+2)- 1 /o=S(</>+^)6==0. Also
This is reduced by replacing ?/ by x+y-x, etc., to Se^-Kr)"^ multiplied
by a factor. On inversion of ($>4"^)~ 1 the rest follows.]
Ex. 2. Find Tp 2 in terms of #, y and z.
Ex. 3. Express the vector in terms of the roots and axes of <
Ex. 4. Prove that
Ex. 5. Prove that p=(<^+%)(^+r(^+y)(^+) is the equation of
a tangent to the curve of intersection of the quadrics y and z ; u being alone
variable.
[Use (vi.).]
Ex. 6. Prove that p=(</>+^r(^+.y)(<#>+ is the equation of the
surface of centres of the quadric x the locus of the principal centres of
curvature when y and z vary. (See Art. 82, Ex. 4.)
Ex. 7. Find the lengths of the principal radii of curvature in terms of
x, y and z.
Ex. 8. The imaginary right line, t variable,
is an umbilical generator of the quadric x.
126 CONFOCAL QUADRICS. [CHAP. ix.
[It is evidently a generator of the quadric, and parallel to a line to a
circular point at infinity for T(<+.r)*e=0. That is, it is one of the eight
generators through the four points in which the imaginary circle at infinity
cuts the quadric. But the tangent plane at an umbilic cuts the surface in a
point circle or a pair of these imaginary generators. See Art. 67, Ex. 1, p. 96.]
Ex. 9. Find the locus of a point through which two of the three inter-
secting confocals coincide. Show that it is a developable surface generated
by the tangent lines to the curve
p =
[This is the locus of the umbilical generators of the system, or the circum-
scribing developable.]
Ex. 10. The focal conies are double curves on this developable.
[Put t equal g ly g 2 or ffa ^ * ne equation of Ex. 8, and we get a plane
curve in one of the principal planes. For t= g l we have
= -1, Sip=0.
The conic is double on the developable because a double sign is lost owing
to the destruction of the component of the vector normal to the plane.]
Ex. 11. If a is a constant vector, and #, y variable scalars, the equation
represents a quadric surface, <f> being a self -con jugate function.
[Assume the equation of the quadric to be Sp(a^> 2 +6<^>+c)p+l=0, and
determine the constants ct, b and c.]
Ex. 12. Prove that the imaginary vector e of equation (iv.) satisfies the
relation V-l . e= Ve<j6e.
EXAMPLES TO CHAPTER IX.
Ex. 1. Three right lines through a common point are mutually at right
angles. If the first and second move in the planes SAp=0 and S/xp=Q
respectively, the third describes the cone
Ex. 2. The cone
_
Sap Sp Syp ~
contains the six unit vectors i,j, Tc and, a, ft, y, the vectors of each set being
mutually perpendicular.
Ex. 3. If the cone Sp^>p=0 has three mutually rectangular edges, the
condition m"=Q must be satisfied ; if it touches three mutually rectangular
planes, m'=Q.
Ex. 4. The four cones of revolution which touch the planes
SA,p=0, S/ip=0, Syp=0
are represented by T. Vp^VpSi V^TA,(SXju,v)- 1 =l ;
and the cones of revolution through the three lines
YAp=0, V/ip=0, V^p=0
are represented by T. p~ 1 SpS VjLtvTA,(SA//v)- 1 =l.
ART. 84.] UMBILICAL GENERATORS. 127
Ex. 5. Three points fixed on a line move in given planes. Find the
locus of a fourth point fixed on the line, and show that it is represented by
an equation of the form
T(aV J u,vS Ap + 5 Vi/ASftp + cVA/xS vp) = 1 .
Ex. 6. Interpret the equation
as determining the locus of a point moving in accordance with a certain law
in relation to a given line and a given plane.
Ex. 7. The polar planes of points situated on certain fixed lines cut a
quadric in circles.
Ex. 8. Find the locus of the centre of a sphere which rolls along two
straight wires.
Ex. 9. Determine the locus of the vertex of a right cone standing on a
given ellipse of which a and $ are the principal vector radii.
Ex. 10. A plane cuts a constant volume from a pyramid having its
vertex at the centre of a quadric. Find the locus of the pole of the plane
with respect to the quadric.
Ex. 11. Find a tangent plane to a quadric which along with three
mutually conjugate planes passing through the centre forms a tetrahedron
of minimum volume.
Ex. 12. Find the locus of the point of intersection of three mutually
perpendicular planes each of which touches one of three given confocal
quadrics.
Ex. 13. Find the locus of the foot of the central perpendicular on a
plane through the extremities of three mutually conjugate radii of a quadric,
Ex. 14. Find the locus of intersection of tangent planes at the ex-
tremities of three mutually conjugate radii of a quadric.
Ex. 15. Find the locus of a point whence three mutually perpendicular
tangent lines can be drawn to a quadric.
Ex. 16. Find the locus of a point whence three tangent lines can be,
drawn to a quadric so as to be parallel to three mutually conjugate radii.
Ex. 17. Show that the equation
" P <ffi> _ Q
determines the directions of the radii of the quadric Sp<j>p+ 1=0 which are
most or least inclined to the corresponding normals. Solve this equation.
Ex. 18. Through the extremity of the vector a mutually perpendicular
lines are drawn to cut a quadric. Prove that
where A\ and # 2 are the intercepts on one of the lines.
Ex. 19. From a point on the quadric S/x/)+l=0, the extremity of
the vector a, mutually rectangular lines are drawn to terminate on the
surface. The plane through their extremities passes through the extremity
of the vector 2<ia
a V-.
m
128 QUADRIC SURFACES. [CHAP. ix.
Ex. 20. Find the volume of the frustum of the cone whose vertex is at
the centre of the quadric S/></>+l=0 and whose base is the intersection of
the quadric with the plane SA/>4-l =0.
Ex. 21. If UV# is a fixed vector y, eliminate the scalar t and the
variable part of q from the relation
P^qf
and discuss the locus represented by
Ex. 22. The vectors a, (3 and y being unit and mutually rectangular,
show that the condition that
should be a maximum or minimum is
VafU^a+V^U
where <j> is an arbitrary vector function, and prove that this is equivalent to
(a) Hence derive a theorem concerning the conjugate radii of an ellipsoid.
Ex. 23. Through a variable point Q on a fixed line V (/>-/?) a =0, a plane
is drawn perpendicular to a fixed line (y). Find the locus of points P in the
variable plane for which Top=eTpQ where e is a given scalar.
Ex 24. Show tfhat the section of the cone Sp<j>p=Q by the plane
SAp+l=0 is equal to the section of the quadric Sp<joSA<~ 1 A,+l==0 by the
plane SA/o=0.
Ex. 25. Find the equation of the surface which is generated by trans-
versals of the lines V(/>-/3)a=0, V(/D-/?)a'0 and of the ellipse
p= y 4. y cos t+ y" sin t.
Ex. 26. The envelope of the planes of intersection of the sphere
SSAp- 1 ^! with a variable sphere passing through the origin and having
its centre on the quadric $p<f>p+l=0 is the cone
Ex. 27. From the extremity of the vector 8 which terminates on the
quadric Sp<f>p+l0, a right line is drawn to intersect the vector radius a,
one of three mutually conjugate radii a, /?, y, and to be parallel to the plane
containing the other two. It meets the ellipsoid again at the extremity of
the vector - 8 - 2aSS<a ; and the plane SXp-h 1 =0 which passes through the
three points thus determined by the three radii is given by
Ex. 28, Show that
is the locus of the centres of sections of the quadric S/></o+l=0 made by
planes through the intersection of the planes SA/>+l=0, 8X^+1=0; and
-discuss the nature of the curve.
ART. 84.] EXAMPLES. 129
Ex. 29. Show that the surface represented by the equation
may be generated by the intersection of two perpendicular planes each of
which contains a fixed line.
Ex. 30. Prove that the foci of central sections of the quadric Spd>o +1=0
generate the surface
P 2 (V>4>P) 2 Q
^ ~" a
Ex. 31. The envelope of a sphere which passes through the centre of a
quadric and which cuts it in a pair of circles is a quartic surface touching
the quadric along a sphero-conic.
Ex. 32. Quadrics similar to Spfy>+l=0 are described on a system of
parallel chords of S/D</>+l=0 as diameters. Prove that the envelope of
these quadrics is also a quadric, and find its equation.
Ex. 33. Prove that v 2 v , 2m'
2/> w 2 -SaS/> n + =
where p n is the vector to the foot of a normal from the extremity of the
vector a to the surface S/><jfy) + l=0 and where mf and m are the second and
third invariants of the function <.
Ex. 34. If a right line cuts a quadric at the angles 9 and ff, show that
sin 0_sm8 f
where p and p' are the central perpendiculars on the tangent planes at the
points of intersection.
Ex. 35. If n is the length of the chord which is normal to a quadric at
the extremity of p,
Ex. 36. Pairs of mutually rectangular tangent planes are drawn through
the extremity of the vector a to the quadric surface S/></>+l=0; prove
that the locus of their intersection is
and show that this equation may be reduced to
Ex. 37. The sum of the products of the perpendiculars from the two
extremities of three mutually conjugate diameters on any tangent plane to
a quadric is twice the square of the central perpendicular on the tangent
plane.
Ex, 38. In terms of the vectors r=p 2 -p l9 <r=Yp 1 /)2) show that the
equation
represents the chords of the quadric S/></>+l=0 which enjoy the property
that the normals at their extremities intersect.
Ex. 39. The locus of the centres of chords at whose extremities the
normals intersect and which are parallel to a fixed direction r is the right
line S/O<T=O, S/yn/n-=0.
J.Q. I
130 QUADRIC SURFACES. [CHAP. ix.
Ex. 40. Prove that the squared radii of the circular sections of the
quadric ^/) 2 +2SA/)Sftp-i-l=0 which pass through the extremity of the
vector a are
where #, <j and g" are the latent roots of the linear function determining the
quadric. Interpret these results.
Ex. 41. Determine the spheres cut in diametral planes by a quadric.
Ex. 42. If planes through an edge (p) of the cone S/x/>=0 and through
the vectors a and ft respectively meet the cone again in edges coplanar with
the vector y, show that
S (pSa<f>a - 2aSp<f>a)(pSft<t>ft - 20S/wfr8) y = 0,
and reduce this by the aid of the equation of the cone to
= 0.
Ex 43. Using the notation of Art. 38, p. 42, show that if a translation
represented by the vector w will carry the tetrahedron ABCD so that it
becomes inscribed to the quadric Sp<f>p+ 1 = 0, we shall have
Ex. 44. It is required to place a pair of tetrahedra ABCD and AVc'o' so
that their vertices may be corresponding points on a pair of confocal
quadrics. (Robert Russell.)
(a) A quaternion statement of this problem is to determine a self-
conjugate function <, a scalar ^, a quaternion q and a pair of vectors K and
K so that the conditions
<"*(/>- K) =(<>+ u) (qp'q~ l - *')=# writ vector
may be satisfied when p and ' terminate at corresponding vertices of the
tetrahedra in their initial positions.
(b) If < is the linear vector function defined by the relations
we find that u$~ l =$'<]>- 1, and q( ) 0- 1 =(<'<)" V-
(c) Also in the notation of Art. 38, K and u are given by
Ex. 45. A plane mirror (normal v) is moved so as to reflect the light
from a star in a fixed direction (8). Show that if y is the unit vector
towards the celestial pole, <r the unit vector towards the star at the time
jf=0, the vector v must describe the cone represented by
& 2
HKy^cry' + S) or v 2 Sy(cr+8) = 2SvySi/5.
(a) Show that the vector
-i I _? ? * *
y T Ay 7r .y" r <ry ;r .y~Xy*
is independent of t provided the vector A. satisfied a certain condition of
perpendicularity, and interpret
CHAPTER 5.
GEOMETRY OF CURVES AND SURFACES.
(i) Metrical Properties of Curves.
ART 85. Supposing that from each point of a curve a vector
>; is drawn, variable with the position of the point, let us
consider the rate of rotation requisite to produce the change of
direction of the vectors 77 as we pass along the curve. In the
figure P and P 7 are any two points on the curve, and the vector
PH = U77 is a unit vector along the emanant vector 77 drawn
from P, while P'H' = 1177' is a unit vector along the emanant 77'
drawn from P'. The vector PH" is drawn equal to P'H'.
^-""H'
U^ H
FIG. 27.
In the limit the quaternion
W-TLy HH"
is a vector perpendicular to 77 and to rf so that rotation round
it from rj to vf is positive, the angle of the quaternion (the
exterior angle at H) being ultimately equal to a right angle.
The tensor of this vector is ultimately equal to the ratio of the
circular measure of the angle HPH" (the angle between 37 and rf)
to the arc of the curve, and thus the vector represents in
magnitude and direction the rate of rotation in question. In
132 METRICAL PROPERTIES OF CURVES. [CHAP. x.
terms of the differential of IT?? and the corresponding differential
of />( = OP), the vector of rotation is
the second form of the expression for the vector being deduced
from (iv.), Art. 53, p. 68, and the third form resulting from the
consideration that
If, in particular, we replace the vector YI by dp, a vector
tangential to the curve, we have for the vector of rotation of
the tangent, or the vector cwrvatiire at P,
_vdpdv
Tdp 3
for in accordance with the foregoing this vector represents in
magnitude and direction the rate of bending of the curve at the
point P, the bending taking place in the plane through P at right
angles to this vector.*
In the case of a plane curve this vector curvature is always
parallel to a fixed direction that of the perpendicular to the
plane, but in the general case the direction of the vector is
continually changing. The plane through P to which it is
perpendicular, or the plane of the bending at P, is the osculating
plane of the curve at P.
To investigate the rate of rotation of the osculating plane as
we pass along the curve, or, what is equivalent, the rate of
rotation of the normal UVdpd 2 p to that plane (compare the
third form of (in.)), we have by (11.),
dTJVdpd 2 p _,
since dVdpd 2 p = Vdpd 3 p. This is the vector torsion of the curve
at P. It gives in magnitude and direction the rate of rotation
of the osculating plane, and we see (what is geometrically
obvious) that the osculating 'plane rotates about the tangent
line (Udp).
* The phrases vector curvature and vector torsion correspond to Hamilton's vector
of curvature and vector of second curvature. We shall see what advantage results
from considering an angular velocity to be a vector on the plan of this article,
and the present case is quite analogous. It is easier in Quaternions to represent
the primary characteristics of a curve, the curvature and the torsion, by vectors
than to represent the somewhat artificial and indirect conception of an osculating
circle or radius of torsion. The theory of emanant lines has been worked out by
Hamilton (Elements of Quaternions, Art. 396).
ART. 85.] VECTOR CURVATURE AND TORSION. 133
The vector curvature and the vector torsion may be com-
pounded into a single rate of rotation
which may perhaps be called the vector twist of the curve. This
rotation produces the same effect on the tangent line and on the
osculating plane as the vector curvature and the vector torsion
respectively, for the former vector is at right angles to the
osculating plane and the latter is parallel to the tangent line,
and we do not here consider the rotation of the osculating plane
in its plane or the rotation of the tangent line round itself.
If the equation of the curve is given in the form considered in
Art. 48, that is if p is given as a function of a parameter t, the
expression (v.) may be written in the form
(VL)
where />', p" and p"' are the successive deriveds of p with respect
to the parameter.
If the arc of the curve is taken as the independent variable,
and if p v p 2 > /3 e ^ c -> denote the successive deriveds of p with
respect to the arc, the relations (compare Art. 48, p. 63)
= 0, etc., .......... (VIL)
found by equating to zero the successive deriveds of Tp 1? serve
to simplify the various formulae. Thus (v.) becomes
(VHI.)
Ex. 1. Show how to connect the deriveds of p taken with respect to t
and with respect to s.
Ex. 2. Show that the tangent line and the osculating plane of any curve
may be written respectively in the forms,
x and y being variable scalars.
Ex. 3. The tangent line and osculating plane of the twisted cubic
&
may be expressed by
respectively, a being a constant vector and < a given linear vector function.
Ex. 4. Calculate the vector o> for the helix
W = a(i cos t +j sin t) + kit,
i,j and Ic being mutually rectangular unit vectors.
134 METRICAL PROPERTIES OF CURVES. [CHAP. x.
' Ex. 5. Find the centre of the osculating circle of a curve.
[The vector to the centre from the point on the curve has the same
direction as Vp'p'T//- 3 . Up', and its tensor is the reciprocal of that of this
vector.]
ART. 86. The important relations (n.) and (iv.) of the last
article enable us to reduce every affection of the curve to a
function of the unit vectors
(I.)
of the scalars
and of the deriveds of these scalars with respect to the arc.
We notice first that a, ft and y form a mutually rectangular
unit system so that a/3 y, /3y = a, ya = /3. The scalars a and
c x are the ordinary scalar torsion and curvature respectively,
and j partly for the sake of symmetry we regard them as the
deriveds -5-, -=- of two angles a and c. The angle a is the total
angle through which the osculating plane has turned about the
tangent line in passing from some initial point P on the curve
to the point P. In like manner c is the total or integrated angle
through which the tangent line has turned in the osculating
plane from P to P. The vector a is along the tangent, f$ along
the principal normal and y along the binormal to the curve.
Denoting still deriveds with respect to the arc s by sufSxes,
the fundamental formulae, (II.) and (iv.) of the last article, give
in accordance with (i.) and (n.) of the present, the simple relations
(ill.)
or a^Cj/3, ) 8 1 = a 1 y-c 1 a, y^-fl^a, .............. (IV.)
or simply ^-Voty, ................................ (v.)
if 77 stands for a, /3 or y.
The formulae in a and y are translations of the formulae of
the last article. The formula in /3 is derived from these by aid
of the relation /3 = y.
To express the successive deriveds, with respect to the arc, of
the vector to any point on the curve in terms of a, /3, y and of the
scalars a 1} ^ and the deriveds a 2 , c 2 , etc., of these scalars, we have
ART. 87.] KINEMATICAL METHOD 135
and in general we shall find the n tli derived to be of the form
f> n = aA n + {3B n + yC n , ..................... (VII.)
where A n , B n and O n are certain scalars (not the n th deriveds of
scalars A, B, C, however). We may remark that the deriveds
of highest order of c x and a^ occur in p n in the term /Scn+ya^.^,
as we see from (vi.).
Thus, as we have asserted, every affection of the curve may be
expressed in terms of a, /3, y of a^ and c v and of the deriveds of
these scalars. (See Appendix. Elements, Vol. ii.)
ABT. 87. The developables connected with the curve may all
be investigated in one common way.
The vector r\ and the scalar e being in some way variable with
a point on a curve, a plane of any developable connected with
the curve is expressible by an equation of the form
S(w-/o)>7 = e, .............................. (L)
57 being the variable vector to a point in the plane, and p being
the vector to the point P on the curve to which the plane corre-
sponds. The equation of a successive plane is of the form
(S(v-f>)v-e) = Q, ............ (n.)
e, YI and p being regarded as functions of the arc s, but 57 being
independent of s. Thus two successive planes intersect in the
line of intersection of the first plane and of the plane determined
by equating to zero its derived with respect to & The inter-
section of the plane (i.) and its consecutive is accordingly the
line common to (i.) and to the plane
(m.)
J7 X and 0-L being the first deriveds of 77 and e.
This line of the developable is also given by the vector
equation (Art. 35 (i.), p. 39),
where is a variable parameter.
In the same way, equating to zero the second derived of (i.)
1 with respect to s,
(v.)
and combining this with (m.) and (L), we have the point of
intersection of three successive planes of the developable,
CT==
' ^
136 METRICAL PEOPEKTIES OF CURVES. [CHAP, x,
This point is on the cuspidal edge of the developable, and it
corresponds to the point P on the curve. More generally if in
(vi.) we allow the arc to vary, we have the equation of the
cuspidal edge of the developable.
In particular, the polar developable corresponds to rj = a, e = ;
while ?7 = /3, = gives the rectifying developable; and 77 = y,
e=0 is the tangent line developable. It is shorter in many cases
to treat the developables ab initio rather than to substitute in
the general formulae (iv.) and (vi.).
Ex. 1. The vectors from a point on the curve to the centres of the
osculating circle and sphere are respectively
5 and 2+yi.I
<^ c : ' da Cj
[These expressions follow from consideration of the polar developable.
Or the first is geometrically obvious, and it is also evident that the centre
of spherical curvature lies on the polar line, T=p-f -4-#y, which is by
geometry the locus of points equidistant from three consecutive points on
the curve. To determine x we may express that txT is the vector to a point
which is momentarily stationary as we pass along the curve. Thus
and therefore .-!.*
We must remember that x is not here a function of s, &r is some small
scalar. See the next example.]
Ex. 2. For a spherical curve
[In this case we can determine as so that the vector in the last example
terminates at a fixed point in the centre of the sphere containing the curve,
and now 8x : da is the derived of x with respect to s, so that
_ ===
Cj ds ds d* \OjdA
The method here employed is often useful. The condition may also be
found by expressing that the vector to the centre of spherical curvature
terminates at a fixed point. The condition is momentarily true (not an
identity) if five consecutive points lie on a sphere.]
Ex. 3. Prove that the rectifying line is V(CT-/o)o>=0, and that the
cuspidal edge of the rectifying developable is T=p - - : -$
tf d
[The rectifying plane S(57-p)/2=0 through the tangent line and at right
angles to the osculating plane, generates this developable.]
Ex. 4. The curve is a geodesic on the rectifying developable.
[Prove that the angles of the quaternions
(co+do)):a and (w+dcu) :(a-t-da)
are equal to the second order of small quantities, and hence show that
when the developable is flattened out the curve becomes a right line so
bhat it is a line of shortest distance (or a geodesic) on the developable ] '
ABT. 88.] THE DEVELOPABLES. 137
Ex. 5. If the ratio of curvature (scalar) to torsion is constant, the curve
is a geodesic on a cylinder.
[If To. cos .?"=!, To)sin-fi r =c 1 , the angle H is here constant, and
equations (m.), Art. 86, give d(a cos ff+y sin #)=(), or on integration U(o=&,
a constant vector. The rectifying developable is therefore a cylinder.]
Ex. 6. Show how to determine the curves for which the ratio of
curvature to torsion is constant.
[By the last example we have c^ = ya sin H . Too = Va . Ta>. If dt = Ta> . d$,
we have, on changing the variable from s to tf, a'V&a, and on
differentiating,
or -pa
The integral of this equation is a cos J 2 r =.Xcos+ju l sin, and as we
must have S^a=-cosjS r and Ta=l, it appears that A and /z- must be
perpendicular to one another and to k, and that their tensors must be equal
tosinJT". Thus
a = k cos H+ sin ff(i cos t +j sin t),
and on integrating again
57 SB Jads = p Q + ks cos H+ sin H. J (i cos t +j sin t)d.s,
where /> is a vector constant of integration.]
Ex. 7. Find the conditions that the unit vectors (a, /3, y) of one curve
may remain constantly inclined to those (a', /?', y') at corresponding points
of another.
[We must have o>ds=a>'ds', or ada+ydc=a'da'+y'dc / . Hence either
ft' \\ /3 or else da : dc= -Sy/3' : Sa/3'= const. In the second case both curves
are geodesies on cylinders. In tne first, if a 7 makes the angle u with a,
y makes the same angle with y (the four vectors being coplanar), and
H=u+ H'. In other words,
da = cos u . da' sin u . dc 7 , dc = sin u . da' + cos u . dc'.]
Ex. 8. Find the unit vectors for the locus of centres of spherical
curvature, and show that they remain constantly inclined to those of the
given curve.
Ex. 9. The vectors o and />' are drawn from a centre of reciprocation to
a point on a curve and to the corresponding point on the cuspidal edge of
the developable into which the curve reciprocates, prove that
where K is the radius of reciprocation and where y and y' are unit vectors
normal to the osculating planes at p and />'.
(a) Compare the curvatures and torsions of the two curves.
Ex. 10. Compare the unit vectors for a curve and its inverse.
(ii) Ruled Surfaces.
ART. 88. Having showed in the last article how to determine
the surfaces generated by planes connected with the curve, we
shall now consider the surfaces generated by the emanant line
(compare Art. 85, p 131)
V(tff-p)* = .............................. (L)
138 RULED SURFACES. [CHAP. x.
Reverting to p. 40, Art. 36 (v.), the shortest vector QQ' from
the line to any other line V(&p')rj'=Q is
and OQ= /
Putting in these />'=/>+ dp, ij'=>,+d^ and proceeding to the
limit having divided Q'Q by T(/>' /o), we find
QQ' vj^Udp <JJd
pt '
by Art. 85 (il.); and neglecting a vanishing term in the
expression for OQ,
the various transformations being easy consequences of the
formula just cited, and p being a scalar defined by
d
The vector PL represents the rate of translation of the emanant
line as it passes through successive positions, this vector being
the ratio of the shortest distance between consecutive positions'
to the arc ds of the curve. In other words, the emanant m$y
be supposed to pass from one position to a consecutive in virtue
of a rotation ids about the shortest distance QQ' coupled with a
translation QQ' = p*ds along that shortest distance. Or again-
p is the ratio of the shortest distance to the angle between the
consecutive lines. The quantity p is usually called the para-
meter of distribution of the ruled surface, though the theory of
screws would offer the more suggestive term pitch, because the
transference of the generator from one position to the consecutive
is in the language of the theory of screws effected by a twist
about the screw coaxial with the shortest distance and of pitch p.
The point Q, the extremity of the vector (iv.), is the point of
closest approach of successive generators ; and as s , varies Q
describes the line of striction of the ruled surface. For a
developable, , this coincides with the cuspidal edge, and p
vanishes.
Ex. 1. Prove that the line of striction and the parameter of distribution
of the surface generated by the principal normals of a curve are
Ex. 2. The tangent to the line of striction of this surface is parallel to
mai I 8 d ( Cl ^
a 1 2 +c 1 2 ^ /3 dAa 1 2 +CjV J
and the shortest distance between" consecutive generators is parallel to <o.
ART. 89.] PITCH. LINE OF STEICTION. 139
Ex.3. If i7
prove that tlie condition that the emanant line v\ should generate a
developable is
smZ.d(a+wi)-cossinwdc-0 or sinZ=0.
[By (v.) if J3=0, Scw?d?7=0.]
Ex. 4. Prove that no line except a in the plane of a and fi can generate
a developable ; that the only developables generated by lines in the plane of
a and y are the tangent-line and the rectifying developables ; and that any
line whatever in the plane of ft and y is capable of generating a developable.
[For the plane of a and ft, 1=0 or w=0, and w=0 is impossible if a
varies. For the plane of a and y, I or w = ^. If m=^, we find 7]=U<o
since sin I . da = cos I . dc. If I = ^ , we have a series of developables
and their cuspidal edges are
^=P +
a Q being an arbitrary constant.]
Ex. 5. Prove that the curves
i i
are the evolutes of the curve 57 =p, and that they lie in the polar developable.
Ex. 6. If the emanant is perpendicular to the tangent, prove that
_
Q ~ p + C
where 77 ft cos m + y sin m.
ART. 89. The normal to the ruled surface
& = p+u?i ................................ (i.)
at any point 57 is parallel to
v^Vyfip+udy), , ......................... (II.)
this vector being perpendicular to every tangential vector
&& = dp+udr)+rjdu ........................ (m.)
The tangent plane is
and as it generally involves u } it varies from point to point along
the generator. Moreover, since it involves u linearly, the an-
harmonic of four tangent planes is equal to the anharmonic of
the four corresponding normal vectors (n.), or of the four cor-
responding points of contact (i.), (Art. 37, p. 41).
Expressing that the tangent planes at two points u and <w/ on
the same generator are perpendicular, we have a relation
= , or
140 RULED SURFACES. [CHAP. x.
which determines an involution between the corresponding points
u and u'. This may be thrown into the form
because (SXyu'^-T^/x- 1 )^ -^(VX^- 1 ) 2 . Comparing with
equation (iv.) of the last article, it appears that the point Q in
which the generator meets the line of striction is the centre of
the involution, and that the foci are imaginary. If C and C'
are the two points u and u', it is not difficult to see that this
equation (vi.) is equivalent to
QC.QCr^+j9 2 , ........................... (VII.)
QC and QC' being vectors, and because their product is positive,
they must be oppositely directed. That the quantity on the
right in (vi.) reduces to T?;" 2 ^ 2 follows most easily by taking
the arc as the independent variable, and then
by (v.) of the last article.
Ex. 1. If the tangent planes of a ruled surface touch the surface all
along the generators, the surface must be a developable or a cylinder.
[The direction of the normal must be independent of u. This requires
d77||77, that is, dU^O, or else dp||7/, or the line is a tangent to the curve
<*/>]
Ex. 2. If for any point p=Q the tangent plane touches all along the
generator.
[A generator of this kind is said to be torsal. A ruled surface has in
general a definite number of torsal generators.]
Ex. 3. The point Q being on the line of striction, prove that the tangent
of the angle between the tangent planes at Q and at any point c on the
same generator is
, TCQ
tan A= -.
P
Ex. 4. Prove that the vector velocities of the points c and c' are at right
angles, and compare their magnitudes.
[The vector velocity of c is t(QC+jo). See Art. 88.]
Ex. 5. Prove that the vector to a point on the line of striction of the
quadric Sp<j>p 4-1 = 0, and the corresponding parameter of distribution are
respectively
where Sw^ = 0.
[See 4f t /, 88 - To re <*uce we may take 77 to be a unit vector so that
S^'=0, 877^77=0 as well as 877^7=0.]
ART. 90.] NORMAL AND GEODESIC CURVATURE. 141
(iii) Curvature of Surfaces.
cting a cu
ature of th
d .Ud . k-*Vkp _ V . Ar J
AET. 90. Projecting a curve on any plane, normal to the fixed
vector k, the curvature of the projection is (Art. 85 (in.), p. 132)
or the curvature of the projection is the projection of the cur-
vature into the cube of the cosecant of the angle between the
tangent to the curve and the normal to the plane of projection.
If the plane of projection is parallel to the tangent, the pro-
jection of the curvature is the curvature of the projection.
Resolving the vector curvature -of a curve traced on a surface
into its components perpendicular to and along the normal v,
we have
dUd/> _ 1V dUdo _ 1Q dUd
_^, IVv-g^+y 18,-g-
and since Si/dp = 0, the first component is, by what we have just
proved, the curvature of the projection of the curve on 'the
normal plane (jLi/dp) to the surface through the tangent line, and
the second is the curvature of the projection on the tangent plane.
Remembering that Svdp = 0, and that its derived is also zero,
or Syd 2 p = Sdvdp, the first component admits of the trans-
formations
1Tr dUdp ,,, , , Q , TT , Syd 2 p Sdydp /TTrN '
v - l \v T~ r = v \dp) 1 fej/dUd/)=j -- ^- r -= -- - Ji__.(ni.)
dp r r dp. v. Lap dp.v.Ldp x J
The last of these shows that the component is the same for all
curves traced on the surface, provided they have a common
tangent line dp, dv being a linear function of dp ; and thus in
particular it is the curvature of the normal section of the surface
through dp. This is Meusnier's theorem the magnitude of the
curvature of the normal section is that of the oblique section
into the cosine of the angle between their planes.
The second component is, as we have already shown, the cur-
vature of the projection of the curve on the tangent plane, or it
is the rate of bending of the curve round the normal (or in the
tangent plane). It vanishes for a geodesic the straightest curve
on the surface between a pair of points for such a curve can
have no component of bending in the tangent plane ; and it is
called the geodesic curvature of the curve. The differential
equation of a geodesic is therefore
S>xlpdUdp = 0, or Si/dpd 2 p = Q ................. (iv.)
142 . CUKYATUKE OF SURFACES. [CHAP. x.
The normals to the surface along the curve trace out a ruled
surface, and by Art. 88 the equation of the line of striction and
the value of 'the parameter of distribution are
The tangent planes along the curve generate a developable.
This and its cuspidal edge are respectively represented by
/ v
(vi.)
ART. 91. If fp is any scalar function of />, and if we write
dfp nSvdp, di/ = 0d/>, ..................... (l.)
the function <f> is self-conjugate when n is independent of p or
when it is a function* of fp.
Let dp and d'p be any two independent differentials of p so that
d'dp-dd'p, d'dfp = dd'fp ...................... (ii.)
We find on expansion by (l.) if dn = Svdp,
d f dfp = nSc/>d'pdp + nSvd'dp + Scrd'pSvdp,
dd f fp = nS(f>dpd'p + nSvdd'p + So-dpSvd'p ;
and by (n.) these expressions give
Sd/>(w0d> + vSo-d'p) = Sd'p(n<f>dp + vSo-dp) .......... (in.)
The function n<f>& + v&(ri3 is therefore self-conjugate; and if n
is constant so that a- is zero, or if it is a function of fp so that
<r || v, the function is self -conjugate likewise. We also observe
that if e is the spin- vector of 0,
27ie4-"VW = and Si/e = ................... (iv.)
This scalar condition is in fact the condition that Svdp =
should lead to an integral fp = const.
If the equation of a surface is given in the form //> = const.,
the differential vanishes if dp is any tangential vector at the
extremity of the vector p, and the vector v is parallel to the
normal.
ART. 92. In applying the results of the last article to the
study of surfaces, we shall leave TV arbitrary, and shall write
= +Ve. The spin- vector e disappears automatically from
Sdpdi/ = Sdp$dp = Sdp0 d/>, whatever vector dp may be, and it
also disappears from Vi/dv=VX0 +Ve) dp, because in this case
Svdp = and also Si/e=0 by (iv.) of the last article, so that
Vv v edp = 0. Thus we have
. (i.)
* This is included in a more general theorem (Art. 60, p. 80).
ART. 92.] PRINCIPAL CURVATURES. 143
Writing G for the magnitude* of the curvature of the normal
section parallel to dp,
and it follows at once by Art. 73, p. 107, that is the inverse
square of the radius of the conic
0, ..................... (ill.)
which is parallel to dp. It is also evident from (i.) that Vvdv is
parallel to the radius of this conic conjugate to dp.
Remembering that the function <pl is independent of dp, al-
though it involves p in its constitution, we may for any point on
the surface regard <f> as constant, and we may apply the formulae
of Art. 75 to calculate the directions of the principal axes of the
conic (in.). The inverse squares of the principal radii of the
conic are the principal curvatures (G 1 and <7 2 ) of the surface, and
are the roots of the quadratic
, or
and unit vectors (r x and r 2 ) along the principal axes are deter-
mined by
The three vectors r v r 2 and Ui/ form a mutually rectangular
unit vector system, and we suppose the directions chosen so that
T 1 To = "Ul/.
Writing also
the expression for the curvature (11.) of the normal section
reduces to
(7=a i cos 2 Z+C r 2 sin 2 ^ ...................... (vn.)
by (l.) we also have
Vvdv = (T 2 X cos I - Tl <7 2 sin QT^Tdp, ........... (vm.)
since Vr^r^p^Udp = T-fir^oUdp T 2 ST 1 Udp
= T 1 Sr 2 ^> r 2 sin I T 2 Sr 1 T 1 cos I ;
and the vector OQ to the point of closest approach of consecutive
normals along dp and the scalar p (Art. 90, (v.)) } assume the
forms
TT 0^008^+ g 8 ain? (fi-G^ainteosE . .
u<4-p u ^-2222 f-* ^ }
* It is not hard to see by considering the sense of rotation that if we suppose
C to be positive for a surface like an ellipsoid, the sign selected in (n.) requires v
to be drawn on. the r cwwex side. Of course there is no ambiguity about th&
vector curvature.
144 CURVATURE OF SURFACES. [CHAP. x.
The quadric
(x.)
in which BT is variable and p constant, has complete contact of
the second order with the surface. We have in fact at the point
trr=p, = SdG70 dsT+SKi 2 57, where dsr and d 2 sr are differentials
of tar as terminating on the quadric, and this is also true for
differentials of the vector to a point terminating on the surface.
The equation of the quadric may also be written in the form
S(w-p+0 -M#o( By -p+*o" 1 ') SSB S^- 1 i;, ......... (xi.)
and it is not difficult to prove that the principal curvatures are
the parameters of the conf ocal quadrics
sfa-p+^-tytyo- 1 -^- 1 ^^
which pass through the extremity of p. The subject will be
resumed in Art. 156, p. 295.
AET. 93. The equation of the normal to a surface at the
point p being
or=/> a??, ............................... (i.)
to find the condition that two successive normals should intersect,
we express that the extremity of rs is momentarily stationary
we have
........... (n.)
where dx is some small scalar if dp is small (see Art. 87, Ex. 1).
The condition of intersection is therefore
Sd/>*li/=0, ............................ (in.)
and this is the differential equation of the lines of curvature.
Moreover we have from (n.)
dp ||(1 OJ0)- 1 !/, where Si/(l x<p)~ l v=Q, ......... (iv.)
because Si/dp = 0, and from these equations we can find the
directions of the lines of curvature and the principal curvatures
G r i=0r 1 Tir 1 , (7 2 =a; 2 - 1 Ti/- 1 if x l and x 2 are the roots of the
quadratic.
More directly, we have for the vectors to the centres of
curvature,
&i=*p-a i - l 'Uv, &2 = p-C 2 - l 'Uv, ............... (v.)
and if d x /> and dg/o are tangential to these lines,
d 1/ ) = C 1 - 1 d 1 Ui/, d 2/ o = a 2 - 1 d 2 U^; ............... (vi.)
and the measure of curvature, or the product of the principal
curvatures, is
ART. 94.] MEASUEE OF CURVATURE. 145
if dp and d'p are arbitrary tangential vectors, as we may prove
by supposing p and Uv expressed in terms of two parameters.
The interpretation of this remarkable expression is that the
small area determined on a unit sphere by lines drawn through
its centre parallel to the normals round any small contour on
the surface, bears to the area of the small contour a ratio equal
to the product of the principal curvatures.
If we suppose the vector to a point on the surface to be a
function of two parameters t and u, and if we use upper accents
to denote differentiation with respect to t and lower accents for
differentiation with respect to u, we have
dp = p'dt + p,du,
and Tdp 2 = edt* + 2fdtdu+gdu 2 ,
if e=-/>' 2 , /=-Sp>,, flf =-/>,* ................ (vin.)
Writing also v ~ Vp'p, , equation (ir.) becomes
p'dt+p<lutt(vdt+v/Lu) ixte = Q, ............. (ix.)
and according as we eliminate x and dx or dt, du and dx we find
the differential equation of the lines of curvature
dPSp'v'v - dtduS (p' v , + /o/)i/ + dfc 2 Sp,v,v =0, .......... (x.)
or the equation of the principal curvatures (<7=ar ll IV~ 1 )
O a Ti^+OT v S(/>X+' / />/)"'-Sj/V aB .............. ( XI *>
It IB not difficult to see that we obtain for the measure of
curvature the expressions
and that in terms of the deriveds of 6, / and g,
and hence it follows that the measure of curvature is an explicit
function of the quantities e, f and g and of their derived^ so
that the measure of curvature depends only on the expression
(vin.) of the square of a linear element. If then the surface
undergoes any transformation in which the lengths of linear
elements remain unchanged, the measure of curvature preserves
a constant value.
ART. 94. The following kinematical method is often useful in investi-
gating the geometry of a surface. Suppose the vector p to a point on the
surface to be given in terms of two parameters, u and v, and let a unit
vector a be drawn at the extremity of the vector p tangent to the curve
J.Q. K
140
CUBVATIJBE OF SUBFACES. [CHAP. x.
u variable let y be the unit vector along the normal at the same point and
let #=7a be at right angles to both -a tangential vector. These three
variable vectors maybe supposed connected with three fixed unit vectors
*^ e relations
y\ ......................... (l.)
so that the conical rotation represented by q would bring the vectors i,j, k
into parallelism with a, /?, 7. These relations being supposed to hold for all
points on the surface, it follows that q must be a function of u and . It
will be proved in Art. 106, p. 173, that if g is any vector function of u and v,
its differential is expressible in the form,
(n.)
where* u'du+ufiv^ZVdqq- 1 and d()==ad#4-/?dy + ycb if
while of course d involves differentials of a, /3 and y.
We shall write in terms of a, /}, y,
<o / oa'+j8y+yc', w^a^-h/S&v+yc,, ................... (IIL)
so that equation (v.), Art. 106, is equivalent to
(JFQ (JU, \JV W vv \sw
these being the results of equating coefficients of a, /?, y in the equation
cited :
It will be sufficient for us to confine our attention to the case in which
the curves u and v cut at right angles, so that /J is tangent to v variable,
since a is tangent to u variable. There is, however, no difficulty in taking
the general case. We have then for the orthogonal curves,
dp=Aadu+B/3dv and Tdp*=A*du*+ 2 dv*, ................. (v.)
so that Adu and JBdv are elements of the arcs of these curves. The vector p
being a function of u and 0, we obtain additional relations connecting the
six scalars a', b', c', a & c /? by expressing that
Jferl^>4/^=lfe- ........................
Now, attending to (n.), we have for example, by (m.)j
........... (vn.)
and the differentials of /? and y are obtained by cyclically transposing
a, /J, y, a' 3 5', c', a,, b,, c t . Hence (vi.) at once leads to the three relations
................ (vm.)
obtained by equating the coefficients in
These three relations coupled with (iv.) give all that is necessary for the
investigation.
*Note that Vdqq~ l is not a perfect differential.
ART. 94.] KINEMATICAL METHOD. 147
To ascertain the meaning of the sealars, observe that the vector curvatures
of the curve u variable and v variable are (Art. 86, p. 134)
_
3w* Aa~~ A
so that by what we have shown jl" 1 ^' is the curvature of the normal section
through u variable and A~ l c' is the geodesic curvature of the same curve.
For any curve traced on the surface, if
Ud/>=U(a4dw + /?Mv) = acosZ4-/3smZ, coslds=Adu, sinlds=Bdv, (x.)
the vector curvature is
d.Ud
- yUd/>{(a'sin I - 6'cos Q^
which follows easily on substituting for du and dv in
d . Udp = ( cos Z- a sin QdZ+(/?(c'dw+c,dv) - y(b'du+b,dv)) cos Z
+ (y (a'dw + a v dv) - a (c'du + c,dv) ) sin L
Thus the geodesic curvature depends simply on c', c y , and the rate of
variation of the angle I which the curve makes with u variable. The normal
curvature depends on the four quantities a', a a b', b r The relation (xi.)
includes everything relating to the second differentials of the curve, and if
we write for the curve a'^Udp, y'=U. dUdp.d/r 1 , yV=/3', we may, for
brevity, replace (xi.) by the relation
y=y cosm+ya'sinm, .............................. (x 11 -)
and we may determine the torsion and everything depending on third
differentials by differentiating once more.
Ex. 1. Determine the equations of the lines of curvature, and prove
Gauss's theorem that the measure of curvature depends on differentials of
the line element.
[If <7' and G t are the principal curvatures, p C'~ l y and p C^y are the
vectors to the centres of curvature, and expressing that these are stationary
for the moment, we have
Aadu + j?/jcto - C-\a(b'du + &,dt>) - j3(a'd* + aflat)) 0,
and according as we eliminate the ratio du : dv or G we have the equation of
the lines of curvature, and the equation of the curvatures,
By (iv.) and (vni.) we see that the product of the curvatures is a function
of A, B and their differential coefficients.]
Ex. 2. Prove that when the curves u and v are lines of curvature,
b'=C'A, a = -CJ3, a'=0, &,=0, c'=-B-^, ^^^
and show that
'dC l _(C'-C / )'dB
148 CUBYATURE OF SURFACES. [CHAP. x.
Ex. 3. If the curves u are geodesies, prove that we may take -4 = 1, and
that in this case
,1 VB
where G is the geodesic curvature of any curve, and I the angle it makes
with the curve u variable.
[Here c'=0, so that A is independent of v, and by a change of the
variable u we may put -4 = 1.]
Ex. 4. Prove that the total curvature of any portion of the surface is
where dS is an element of the surface ; and where I is the angle the bounding
curve makes with the curve u variable, G is the geodesic curvature of the
bounding curve and ds an element of its length.
(a) Examine the case in which the bounding curve is composed of
geodesies.
(iv) Families of Cwwes and Surfaces.
ART. 95. If p = y(tl a, b, c, ...), ........................ (i.)
where 77 is a given function of a variable parameter t and of
certain scalar constants a, 6, c, etc., the equation represents a
family of curves, any particular member of the family being
determined by assigning fixed values to the constants a, b, c t etc.
If there are n constants, the family is said to be w-way, or to be
of the 7i th order.
The curves of the family which touch a given surface or inter-
sect a given curve compose a family of order n 1.
If the given curve is /> = ^(t^), the condition of intersection
*i(t\ a, 6, c, ...) = >7iOi) ....................... (n.)
is equivalent to three scalar equations, so that on elimination of
t and ^ from these, we are left with a scalar equation in the
constants a, 6, c, etc., and thus one of the constants may be
expressed in terms of the remaining n 1.
If the given surface is f(p) = 0, the conditions for contact are
/()=(), o ........................ (m.)
and on elimination of t, a relation connecting the constants is
obtained, so that a family of order n 1 touches the given
surface.
ART. 96. Expressing that an unknown surface /(p) = meets
a curve of the family at the extremity of the vector p in n
consecutive points we have
, etc, ........ (I.)
ART. 97.] FAMILIES OF CURVES AND SURFACES. 149
where the functions 0, 0. 2 , etc., are defined by the relations
dj/ = 0dp, d 2 v ^dV + ^Cdp, dp), etc .............. (n.)
The first of the equations (i.) is equivalent to three scalar
equations, so that the system of equations is equivalent to n+Z
scalar equations. We can from these eliminate t and the n
constants a, 6, c, etc., and the eliminant is a function of />, v, 0,
2 , etc., and is equivalent to the differential equation of surfaces
met in n consecutive points by curves of the family.
In particular, the equation is equivalent to the differential
equation of surfaces generated by curves of the family.
Ex. 1. Find the differential equation of surfaces generated by parallel
lines.
[Here /> = K-Ka, Sva=0, and the equation required is Sva=0, a being
a fixed vector and K being arbitrary.]
Ex. 2. Find the differential equation of cones having a common vertex.
[In this case p = a + ti<, Si'K=0, so that St/(/o-a)=0.]
Ex. 3. Prove that SVav<Vai/=0 is the differential equation of surfaces
generated by lines perpendicular to the fixed vector a.
Ex. 4. The differential equation of surfaces generated by lines which
meet the fixed line V(p-/3)a=0 is SVvV(p-)a. <. YvV(p-/3)a=0.
[If p = K + t\ is a generating line, S(/c-$)aX=0, SvA=0, SAxA,=0.]
Ex. 5. Find the differential equation of ruled surfaces.
[We have Sj'A=0, SA(A 0, SA< 2 (AA)=0, and the equation is obtained
by solving for A, (Art. 74, Ex. 3) from the first and second and substituting
in the third.]
Ex. 6. Find the differential equation of surfaces generated by similar-
and similarly situated curves.
[Here a generating curve is p = K + aa(t) where K and a are constants to
be eliminated and where a(t) is a given function of t.]
Ex. 7. The differential equation of surfaces generated by equal and
similarly situated ellipses is
a and f$ being a pair of conjugate radii.
ART. 97. As in the last article, being given the scalar equation
of a family of surfaces involving n constants,
f(pl a, 6, c, ...) = 0, ...................... ....(i.)
we can determine the differential equation of a surface which at
each point is touched by some member of the family in as many
consecutive points as serve to eliminate the constants.
If only one constant is involved, only one surface is touched
at each point by a member of the family, and that is the envelope
obtained as the locus of intersection of consecutive members by
eliminating the constant a between
150 FAMILIES OF SUBFACES. [CHAP. x.
If two constants are involved, the conditions for contact with
some unknown surface at the point p are
*=^ , /(/>; ,6) = 0, ...................... (m.)
where v is the normal to the unknown surface and j/ the normal
to the surface of the family. The first equation, on elimination
of the unknown scalar x, is equivalent to two scalar equations,
and between these and the second we can eliminate a and 6, and
we obtain the differential equation of the touched surface as a
function of p and v, homogeneous in i/.
When the family contains three parameters, we express that
the surfaces touch at two consecutive points, and we have
V=XV Q , <f>dp=x<t> Q dp+dx.vQ=zQ, f(p] a, b, c) = 0, Sj/d/) = 0. (iv.)
We can eliminate dp and replace the equations by
i/=a?i/ , Sj/to-oj&J-WO, /(/>; a, 6, c) = 0; ......... (v.)
and these equations are equivalent to five scalar equations from
which to eliminate x } a, b and c.
Observe that we find two directions dp for contact according
as we substitute one or other of the values of x given by the
scalar equation (v.) in the second equation (iv.)
It is not hard to see that each additional condition of successive
contact affords one additional scalar equation in x and the
constants. In fact if we attend merely to the new unknowns
d m p and d m x introduced in d m ~ l (<f>dp #0 d/> + d#j/ ) = and
O, we see that they occur in the forms
and when we eliminate the vector d m p, the scalar d m x disappears
also by (v.). The preceding vector condition
d m -\j&p - x<j> dp + d0i/ ) =
serves to eliminate d m ' l x } and so on.
^The conditions of contact at n 1 successive points serve to
eliminate the n constants, and the result is the differential
equation of surfaces touched at each point by some one member
of the family in n 1 successive points. In particular, the
equation is the differential equation of envelopes of the family
obtained by replacing the n constants by arbitrary functions of
a single constant.
When the family of surfaces is given in terms of two para-
meters t and u, /*..,& \ ATTT \
/> = ?7(, u; a, 6, e, ...), ........................ (vi.)
we have v =
and on direct elimination of d, du and dx,
ART. 98.] DIFFERENTIAL EQUATIONS. 151
The next differentiation introduces d% d 2 u and d z x, and these
being eliminated by an equation analogous to (vm.) we use (vn.)
to get rid of dt, du and dec.
Ex. 1. Prove that for the envelopes of a family of spheres,
where K is the vector to the centre of a sphere and R the corresponding
radius.
Ex. 2. The differential equation of envelopes of spheres of constant
radius whose centres lie on a curve on the surf ace //>=0 is/(pTJvjff)=0.
Ex. 3. The differential equation of the envelopes of spheres having their
centres on the ellipse pa cos t+ft sin is
Ex. 4. Find the differential equation of developable surfaces.
Ex. 5. Show how to find the differential equation of the envelopes of a
surface carried parallel to itself.
[Take /o-S+ijfo u).]
Ex. 6. Find the envelopes of a rptated surface.
[Take p-fiMjfotO-r 1 -]
ART. 98. A differential equation of the first order presents
itself in the form
homogeneous in j/. For any variation of p and v subject to this
condition, d.F(p^) = Srdp+Sfjid v =0 ) ................... (n.)
where r and /x, are determinate functions of p and v. If the
equation has a solution, there must be some scalar function of
p, so that d.fp = nSvdp, ........................... (m.)
and for any arbitrary differentials of p t if d7i = S<rdp,
d'dfp = nSd'vdp + nSvd'dp + So-d'pSvdp
= dd'/p = nSdvd'p + nSvdd'p + So-dpSz/d'/),
so that (compare Art. 91)
S^d'v-o-Svd'p+vSo-d'p^dp-nSd'pdv^O', ......... (iv.)
and this general relation must include (n.) as a particular case.
Hence for some differential d'p satisfying Sjxi / / o = 0, we must
have xr=nd'v+vS<rd'p, ^--^d>, ................. (v.)
and from this we have the equivalent of Charpit's equations
O ......................... (vi.)
152 CURVES AND SURFACES. [CHAP. x.
EXAMPLES TO CHAPTER X.
Ex. 1. Determine the equations of the osculating circle and osculating
helix of a curve in terms of the vectors a, /5, y and the scalars a^ and c x
corresponding to the point of contact, and find the deviation of the curve
from the circle or helix.
Ex. 2. Show that the vector to a point on an ellipsoid may be expressed
in the form
p=a,cosu+rsmu where TT=&, SAr=0, TA = 1,
the vectors A and a being constant but r being variable.
(a) A tangential vector is
dp = ( a sin u + r cos u) du + XT sin udt,
and the equation of the tangent plane is
Sv/o = 6 2 SAa where v~ VAr (a sin u - r cos u).
Ex. 3. The differential equation of a geodesic on the quadric S/x/>+l =0
is S</od/>d 2 /o=0.
(a) This equation, which expresses that <p, dp and d 2 p are linearly
connected, may by the aid of the differentials of the equation of the quadric
be replaced by Sd^V , Sdp
and operating by S<dp an integrable relation,
Sdpd'
~
is found which affords the integral
(6) The geometrical interpretation is that PD is constant along the
geodesic, where P is the central perpendicular on the tangent plane and
where J) is the diameter of the quadric parallel to the tangent to the
geodesic. (Compare Ex. 14, p. 287.)
Ex. 4* A unicursal curve of order n is represented by an equation
of the form / n n
(a , ^ a 2 .
and in general this equation may be transformed into
and the curve may be described as the locus of the mean centre of corre-
sponding points on n homographically divided lines.
(a) The equation of the asymptotic tangent parallel to & is
Ex. 5. Find expressions for the curvature and torsion of a line of
curvature on a quadric in terms of the elliptic coordinates of Art. 84
Ex. 6, The ^ vectors p=0(t) to points on a curve are transformed by the
operation of a linear vector function <. Compare the curvature and torsion
at corresponding points.
*See Proc. R.I.A., 3rd Series, Vol. iv., 1897*
ART. 98.] EXAMPLES. 153
Ex. 7. (a) If a, /3, y and S are vectors from a common origin to four
points A, B, c and D, it is always possible to determine four scalars a, b, c
and d, so that aa + bp + cy + dS = 0.
(6) If the sum of these scalars is zero, the four points lie in a plane.
(c) It is also possible to determine a second set of scalars so that
a'cr 1 4- &73' 1 + c'y- 1 + d'S~ l = 0.
(d) If the sum of this new set vanishes, the points lie on a sphere passing
through the origin.
(e) The equation of this sphere may be written in the form
Sp- 1 (/j-y-i + y" 1 ^ 1 + a-i/3- 1 ) = So-ijS-y- 1 .
(/) If it is possible to determine a third set of scalars so that
a"a* + 6"j(J* + c"y 4 + d"8* = 0,
the four vectors are edges of the right circular cone
(g) If the additional condition is imposed that the sum of the scalars of
this third set vanishes, the four points lie on a surface whose equation may
be written
\ being a constant vector.
(h) Discuss briefly the nature of this surface. (Bishop Law's Premium'.)
Ex. 8. The differential equation of surfaces generated by lines of the
complex (Art. 36, Ex. 4, p. 40)
/(o-,r) =
may be found by eliminating cr and r between this equation and
(a) For the linear complex S(ao-+,y8r) 0, the equation is
S . Vv(Vap + ^)<^yv(Va/3 + /:?)=0.
(b) Lines common to the two linear complexes
S(ao-+/3r)=0, S(ycr4-aV)=0,
generate the surfaces whose differential equation is
S . v(Vap + /3)(Vy/o+S)=0.
(c) Find the differential equation of surfaces generated by lines of the
congruency
/( < r,T)=0, S(r+0r)-0.
Ex. 9. If the vector 13 is a given function of a variable unit vector a,
the equation Y (p ~ ^8) a =
represents a congruency of right lines.
(a) If dj@ = <da determine the meaning of the several terms in the equation-
= Pada.
(b) A line of the congruency is intersected by consecutive lines at two
focal points p=/3+xa where x is a root of the quadratic
- 1 a=0, or
being the spin- vector of <f> and < being the self -con jugate part.
154 CURVES AND SURFACES. [CHAP. x.
(c) The points of closest approach of consecutive rays to the ray />=/3-Kra
lie between the extreme points determined by the condition that Suda<Uda
may be a maximum, ana the corresponding values of % are the roots of the
quadratic
or
(d) The vectors of shortest distance at the extreme points between the
ray a and its consecutives are mutually perpendicular ; and if these shortest
vectors are parallel to the unit vectors a and a,, the extreme points are
determined by a^Sa^a, and 3,=Sa'<a'.
(e) If the vectors a, of and a, are in positive order of rotation so that
aV*^ Sa'<K= ~ Sa^a'= - Sea ;
and if the shortest vector at the point corresponding to x makes the angle it,
with of so that \Ja&a = a' cos u -f a,sin u 9
the scalars x and P are connected with #' and x t by the relations,
# / sin 2 u, P Sea + (x, - #') sin u cos u.
Ex. 10. A circle may be represented by means of a pair of vectors (K, A)
since its equations may be thrown into the form
T(/>-/c)=TA, SA(/>-K)=0;
and an equation such as /(*, A)=0,
where / is a general function, may be regarded as representing a family of
circles.
(a) In like manner an equation such as
/( a > A y)=0 where Sa/3=0
represents a family of conies, y being the vector to the centre of one of the
conies and a and $ being its piincipal vector radii. (Compare Ex. 1 1, p. 103.)
Ex. 11. The general surface generated by a variable circle (/c, X) may be
represented by
p=K+Xr where SXr=0, Tr=l,
the vectors K and A. being functions of a single parameter and the auxiliary
vector T being arbitrary so far as the conditions allow.
a) If P is a scalar analogous to the parameter of distribution of a ruled
ce> D^ T j , j \ -a- j rdX-d*
P =d/c+d . AT. Hence dr= -^ - r-,
T JTT A
and because Sdr=0, Srdr=0, we find
Srd/c SrdA
(b) These expressions for P lead to four values of the vector r which
determine points at which neighbouring elements of successive circles
approach most closely or are most widely separated.
(c) If successive circles intersect in one point
rp/'TTj \ \ cj j \ \ . "TTJ \ j \ \ rn
I ( v aAASaAA + V d/cASd/cA) = T . ,
and the vector to the point of intersection is
P=K +
SAdXd/c
ART. 98.] EXAMPLES. 155
(d) If successive circles intersect in two points, the vector just found
becomes indeterminate, and
and when this condition is satisfied, the surface may be generated by the
motion of the sphere,
(e) In the general case, the equation of a normal to the surface is
and when this is expanded we obtain two scalar equations which combined
with the equations of condition enable us to eliminate r, so that we find the
equation of the surface generated by the normals along the circle (K, A.) to be
B(p - K)&K - SXdX S(/>dA - KdA. - d*A) UV(/> - *) A-0.
This surface is of the fourth order, and normals at the extremities of
diameters of the circle intersect in a nodal conic.
CHAPTER XL
STATICS.
ART. 99. If a is the vector to the point of application of a
force which is represented in magnitude and direction by the
vector /3, the moment of the force with respect to the origin is
Va/3 the vector area of the parallelogram determined by a
and /3 ; and the moment about the extremity of the vector y is
V(a y)/3. The force maybe replaced by an equal force /3 at
the origin, and a couple Va/3; or by an equal force /3 at the
extremity of the vector y and a couple V(a y)/3.
For any number of forces, the quaternion quotient of the
resultant vector moment at the origin by the resultant force is
(Elements, Art. 416 (11))
where p = Sq, zrr=Vg; ......... (i.)
and because 2Va/3=p2/3+^2/3=.p2/3+V / >2/3, .............. (11.)
if p is the vector to any point on the line represented by
V P 2^=^E/3=(2/3)~ 1 V.E/32Va^, .............. (m.)
we may replace the system of forces by a force 2/3 acting along
the line (in.) and by a couple _p2/3 having its axis parallel to
that line. This is the reduction to Poinsot's central axis.
The system of forces constitute a wrench upon a screw ;* the
scalar p, which is independent of the origin, is the pitch of the
screw, and the vector & is the perpendicular from the origin on
the axis of the screw Poinsot's central axis.
If the resultant reduces to a single force, p is zero or
SSVa/^S/S)- 1 ^; and if they reduce to a couple 2/3 = and p
is infinite. If the forces equilibrate
2/3 = 0, 2Va/3=0 .......................... (iv.)
*Sir Robert S. Ball, Treatise on the Theory of Screws, Cambridge, 1900.
ART. 99.] WRENCH UPON A SCREW. 157
Hamilton uses a second quaternion
and the scalar of this quaternion is the pitch while the vector
terminates at a point which is independent of the origin the
Hamiltonian centre of the system of forces. This point is
evidently situated on the central axis (in.).
The quaternion a/3 is called by Hamilton the quaternion
moment of the force /3 with respect to the origin. Its vector
part is the moment of the force and its scalar part is minus the
virial. We shall write for any number of forces
2j8 = X, ..................... (vi.)
so that we have
, ............ (vn.)
where jm. is the resultant vector moment at the origin and where
m" is minus the resultant virial at the same point. The plane
of no virial is represented by
-p) j 8 = or SyoA=m"; ............... (vm.)
and Hamilton's centre is obviously the intersection of this plane
and the central axis.
Ex. 1. Vectors (a) are drawn from a variable origin to the points of
application of forces (p). The equation
2Va/3=0
implies equilibrium.
[If the vectors a are drawn from a fixed origin to the points of appli-
cation, we must have separately 2/3=0, 2Voo/3=0 (Elements, Art. 416).]
Ex. 2. Forces act at the vertices of a triangle, in its plane and pro-
portional and perpendicular to the opposite sides. Prove that they are in
equilibrium.
flf a, ft and y are the vectors from a variable origin, the forces are
v(p y), v(y a), v(a~y8) where v is a vector perpendicular to the plane of
the triangle. The moment formed as in the last example vanishes identically
because vav/3=V/3vo, etc.]
Ex. 3. The conditions of equilibrium of a rigid body may be expressed
by the equation 2S/3da = 0,
which contains the principle of virtual velocities (Elements, Art. 416 (17)).
[For any possible small displacement of the body da==S+Vo>a where 8
and (o are arbitrary. Hence 2/3=0, 2Va/3=0.]
Ex. 4. The moment of the force AB about the line CD is six times the
volume of the tetrahedron ABCD divided by the number of units of length,
in CD.
[The vector moment at the point c is V . CA . AB and the component along
CD IS -S(UCD . V . CA. AB) = -S. CD. CA . ABT. CD" 1 .]
158 STATICS. [CHAP. xi.
Ex. 5. A force of unit intensity acts along the line Y(/o~a)/3=0. Its
moment about the line V(/> - a') / = is - S (a - a'J
Ex. 6. If three forces are in equilibrium, they must be in the same
plane,
[Operate on the condition V(p-a)f}+'V(p-a')ff+'V(p-a ir )fr**Q by
S(/>-a) and put />=a' where we find S(a'-a)(a'- a")/3"=0.
Ex. 7. If four forces are in equilibrium, their lines of action are
generators of a hyperboloid.
[One method of proof (Chap. VIII., Ex. 10, p. 103) is to express the four
vector moments Von/? n , etc., in terms of the four forces by means of a linear
vector function, so that Va n /3 n = </?+<> The vector a> is zero because
0, 2/3 n =0, and therefore the equation of a line of action is
(See Art. 79, p. 116,)]
Ex. 8. Resolve a wrench into forces along the edges of a tetrahedron
ABCD.
[If jj, is the moment and A the force of the given wrench at the fixed
origin of vectors o, the moment at the point p is
JU, - V . OP . A = 2#AfiV . PA . AB
where ABJ etc., are scalars proportional to the forces along the edges. Take
the point P at D, and
ft-V.OD. A ^AB- V.DA. AB + #BcV.DB.BC + #cAV.DC.CA
serves to determine three of the unknown scalars. Operate by S . DC and
AB.S.DA.DB.DC=S(JU- V.OD . A)DC, or ZAB.(ABCD)=S.CD .//.-hS.oc.OD. A.]
ART. 100. To reduce a system of forces to two forces, let
JUL and X be the resultant couple at the origin and the resultant
force o the system, and assume
ft = V a p+Va'P, X=/3+/3', (I.)
where /3 and /3' are the two forces and a and a the vectors to
their points of application. Hence
/3' = X-/3, M-V(ct-a')/3 + Va'X; (II.)
and from the form of the second equation, it is obvious that if
two of the unknown vectors a, a', /3 are suitably assumed, the
third may be regarded as the vector to a point on a determinate
line. But a condition must be satisfied, for on operating in turn
by S(X /3) and S(a a) we have
S(X-/3) M = SXa/3 and S(a-a')v = Saa'\, (ill.)
so that if any one of the three unknown vectors is assumed
(say a) the other two may be regarded as terminating on
definite planes. Suitably selecting either ft or a in accordance
with (ra.) (which is a consequence of (n.)), the remaining vector
is constrained by (il.) to terminate on a line.
Ex. 1. A rigid body is acted on by any number of forces. It is required
to equilibrate the body by two forces whose points of application are
situated on given lines.
ART. 101.] BEDUCTION TO TWO FORCES. 159
[If and & are the required forces and Y(/>-a)/3=;0, Y(/>-a')/?'=0 the
equations of the given lines, we have
where x and a! are scalars. Hence
and this equation of condition establishes a homography connecting the
points of application.]
Ex. 2. A framework is composed of rods jointed by smooth hinges.
Three of the rods, A 4 A 1? A 4 A 2 and A 4 A 3 terminate at a point A 4 and are acted
on by given wrenches. Determine the reactions at the joints ; it being
supposed that the three rods are not coplanar.
[Let (fam ^mn) represent the wrench applied to the rod A^A*, the origin
of vectors being taken as base-point, and let fimn be the reaction of the joint
on the rod at the point A. For equilibrium of the rod A 4 A 1?
and putting />=%, this gives
/% - Vo 1 A 41 + Y(a 4 -
or, for some scalar # 41 ,
For equilibrium of the joint A 4) we have /3 4 i+j842+&s= or
^ (ft*, - Yca 4n ) (a 4 - a,,)- 1 = - 2> 4n (a 4 - a*)' 1
and from this vector equation the three scalars # 4n can be found.]
Ex. 3. A rigid body is in equilibrium under the action of an impressed
system of forces (p, A) and the tensions of two strings A'A and B'B attached to-
points A' and B' in the body and to fixed points A and B. Show that the
forces exerted by the strings on the body are represented by
where #, y and t are scalars which may be determined by expressing ^
the lengths of the lines A'A, B'B, A'B' and AB are given, and where a, & a and
ft are the vectors from the base-points to the points A, B, A' and B.
(a) What condition is implied in these equations ?
(6) If a, 6, o and d are the tensors of the vectors A'A, B'B, A'B' and AB,
respectively, show that the scalar t satisfies the equation
ABT. 101. The resultant quaternion moment (Art. 99 (vi.)>
for an arbitrary base-point (the origin of the vectors a) of a
system of forces (/3) acting at points fixed in a rigid body is the
first quaternion invariant of the linear vector function
the first scalar invariant of this function being minus the
resultant virial (m" = 2Sa/3), and double the spin-vector being
the resultant vector moment (/x =
* That is the invariant - 0t* . ' - # ./-#*.*. Compare Art. 67, Ex. 7, P- 97.
160 STATICS. [CHAP. XL
If the forces receive a common conical rotation round their
points of application so that each vector /3 is replaced by qBq~ l ,
the function <j>p changes into <t>(q~ l pq)l and if the body is
rotated so that a becomes qaq~ l } the function becomes q(<f>p)q~\
The results of Art. 70 show that there are four rotations
applicable either to the body or to the forces which render the
function self -conjugate ;* and in this case the resultant is a
single force passing through the origin. These four positions
of the body relative to the forces are called the initial positions.
If X( = 2/3) = 0, the resultant is a couple for all relative
positions. If the forces are in astatic equilibrium, the couple
{as well as the resultant force) must vanish for all rotations ; but
this can only happen when the function vanishes identically
because a function such as q(<pp)q~ l cannot be self-conjugate for
.all quaternions q. Thus the necessary and sufficient conditions
for astatic equilibrium are
= 0, X;=0; (II.)
,and these are equivalent to twelve scalar relations connecting
the forces and the points of application.
In general reduction of the function to a trinomial form
</>p = y 1 SA 1 />+ y 2 SX2/o + y 3 SX 3/ o, X 1 +X 2 +X 3 = X, (m.)
in which X, and X 2 are arbitrarily assumed, corresponds to the
reduction of the system of forces to three forces \ 19 X 2 and X 8
'Ostatically equivalent to the given system; and it is easy to
.see that the points of application of these forces, the extremities
of the vectors 7^ = 0VX 2 X S : SX^gXg, etc., are fixed relatively to
the body and lie in the central plane
SpV/X = m or S/o0 / - 1 X = l (iv.)
Reduction of the function to the standard form of Art. 70 gives
.a particularly simple set of equivalent forces or couples.
The vector <f>\ is obviously fixed in the body, and when the
origin is transferred to the extremity of the vector 0.X" 1 the
linear function (which we continue to denote by 0) corresponding
to this special origin the astatic centre satisfies the condition
0X = (V.)
As one root of <f> is now zero, the function is reducible to the
binomial form, and the auxiliary -^ function is of the type
1/r/> = XS/C/0 (VI.)
where K is a vector fixed in the body. The equation of the
central plane is now S/c/o = 0.
* These are the rotations which convert if, /, k' of the article cited into
+* +j, +&; +, -j, -&; -, +j, -&; or -i, -J, +k. Compare the foot-
.note to the article cited.
ART. 101.] ASTATICS. 161
In addition to the equations (v.) and (vi.) we have
and <//,=<>=Y/cA.; ..................... (vii.)
the first is obvious because p is double the spin-vector and the second follows
from Art. 68 because - <f>p is double the spin- vector of i^- These relations
coupled with the expression
for the moment in terms of the pitch p and the vector rj from the astatic
centre to a point on the central axis of the forces in any position enable us
to deduce all the theorems of astatics. We first remark that the function 4><j>'
is fixed relatively to the body (or to the vectors a) and that the function <j>'<j> is
fixed relatively to the vectors /3 (or to the directions of the forces).
In order to determine the arrangement of the central axes relatively to
the forces, operate on (vm.) by the function <, and by (vn.) we find
(ix.)
so that T<Y7?A.=TY/cA or SY7?A<'<Y?7A==(V/eA) 2 ; ............... (x.)
and therefore relatively to the forces the central axes compose a coaxial family
of similar elliptic cylinders whose linear dimensions are proportional to the
cosine of the inclination (TYTJtfA.) of the central plane to the axes whose
direction (TJX) is of course fixed relatively to the forces.
The arrangement of the central axes in the body is determined by the
equation
(xi.)
obtained by operating on (vm.) by YA. and attending to (vn.). Taking the
tensor
TY9 7 A-T^UA=V(-SUA^ / UX); ..................... (xn.)
and the locus of central axes having a given direction UX relatively to the body
is a right circular cylinder* whose radius is the reciprocal of the parallel
radius of the elliptic cylinder
T<Y>=TA or Sp<W>>=A 2 ........................... (xm.)
To each generator of a cylinder (x.) corresponds one of the cylinders (xn.)
which is traced out by that generator when the forces are rotated round the
vector A. In terms of the vectors cr and r of Art. 36, Ex. 4 (r|| A), we may
replace (xn.) by
TATcr=Tc/>'T, ................................... (xiv.)
and this equation represents a complex of the second order the assemblage
of lines in the body which become central axes by suitable rotation of the forces.
"We shall now determine the pitch corresponding to each central axis.
Operating by <' on (vm.) we have by (vii.)
(xv.)
and operating on this by Scf/A, or SYAju, or SYAY^A. we deduce
j9T^A.2-SA^'Y7 7 A=TA 2 SAC7 ? A. ...................... (XVL)
This equation gives p in terms of the vectors determining the central
axes. Again we obtain an equivalent expression by taking the tensor of
(xv.), and on replacing X by r and Y^A. by cr the result is
^T^T^a^ST^cr+T^^TY/cT 2 . ... ............... (xvn.)
This represents a complex of the second order and the lines common to
the complex (xiv.) compose a congruency of the fourth order and the fourth
J.Q. L
162 STATICS. [CHAP. xi.
class the assemblage of lines in the body which become axes of screws of given
pitch for suitable rotation of the forces*
Since (xvi.) is linear in 77, it represents a plane when the direction of A
is given which cuts the cylinder (xn.) in two axes corresponding to the
given pitch. The plane touches the cylinder if
(xvm.)
and this relation determines the limiting values of the pitch for a given
direction UA.
The function fap corresponding to an arbitrary base-point the extremity
of the vector q-is <^p=<p-7?SAp ................................. (xix!)
because <p is of the form laSftp. The function fafa for this base-point is
fafa'p^Wp-TW.il&np; ........................... (xx.)
and supposing w 2 to be a latent root and a to be a unit vector along the
corresponding axis, it appears on inversion of the function <<'- w 2 that the
latent roots (w 2 , u' 2 , u" 2 ) of fafa are parameters of the quadrics of the con-
focal system (fixed in the body)
Sp(^ / -^ 2 )- 1 pTA 2 =l ............................ (xxi.)
which pass through the extremity of ??, and that the axes (a, a', a") of the
function are the normals to these confocals. Eeduction of fa to the
standard form of Art. 70 gives
(xxn.)
where the unit vectors ft are likewise mutually perpendicular so that the
system of forces may be replaced by A acting at the extremity of ?/ and by
three couples (such as that due to the unit force +6 acting at the extremity
of tj + ^ua and (3 acting at the extremity of 77 f ua) whose arms (wx, wV,
u"a") are mutually perpendicular as well as the forces (ft, ft\ ft").
The parameters of the confocals (xxi.) touched by an arbitrary line (cr, r)
are the roots of the quadratic equation (Art. 83, Ex! 2, p. 124).
where M" is the first invariant of <<', observing that in general the
i/r function of <<' is ^r'-^r ; or of the equation
w^T 2 -^ 2 (l/'TT 2 -T^V 2 +Ta^A 2 )+T^T 2 +T^o- 2 TA 2 ==0 ; ..... (xxm.)
and when the line belongs to the complex of central axes (xiv.) the equation
reduces by (xvn.) without much trouble to
(xxiv.)
where Jf' (=TK 2 TA 2 ) is the second invariant of <<' or the first of i^/r. This
shows that the central axes touch confocals having the sum of their parameters
constant and equal to W \ and in particular we have Minding's theorem for
^?=0 that the lines of action of single force resultants intersect the focal conies
of the system (xxi.) since the parameters of the touched confocals are in this
case the finite latent roots of <$><fi and the focal conies obviously correspond
to these parameters. The theorem respecting the constant sum of para-
*The former equation (xvi.) in terms of r and a- is
= TXTrS/co-
#T0V 2 - ST00V = TXTrS/co- ;
and on rationalization this is seen to represent a complex of the fourth order, and
it may be shown that coupled with (xrv.) it reduces to (xvn.) affected by the factor
Ttf/A
ART. 102.] ASTATICS. 163
meters is otherwise deducible from Art. 83, Ex. 3, for the cone of lines of
the complex (xiv.) through the extremity of the vector 7] is expressible in
the form Sr(^' - T^TA 2 - ^TA 2 . 8^=0 ................... (xxv.)
Moreover (Art. 83 (x.)) this is the reciprocal of the tangent cone to the
confocal (xxi.) whose parameter is ^ 3 =iyTA 2 . According as the tangent
cone becomes more and more obtuse by variation of the vector ?? and finally
becomes a tangent plane, the reciprocal cone becomes more and more acute
and finally coincides with the normal to the quadric, and the locus of such
points is the surface
This surface is a quartic analogous to Fresnel's wave-surface, and its
equation may be reduced to the form
TY/o? ___ T/c
""
remembering that <'*=(). In this form it is apparent that the surface
consists of a system of circles concentric with the astatic centre, coplanar
with the vector K and of radius proportional to that of the elliptic cylinder
(xni.) which is parallel to the ractius in the central plane. For points inside
this surface the cones of axes are imaginary.
The boundary of the region containing the feet of central perpendiculars
on the axes has been investigated by Tait (Quaternions, Art. 403).
Expressing that Tq is a maximum when U^ is given and when r is
subject to the conditions (xxv.)
the equation of the boundary is found to be
S^O^'-T^TA-V^O; ........................... (xxvm.)
and this represents a surface of the sixth order analogous to the inverse
of a Fresnel's wave-surface, and on expansion it aifords a quadratic in T^ 2
corresponding to any given value of U?/ whose roots are the limiting values
of the squares of the perpendiculars.
Ex. If vectors are drawn in the body from an arbitrary base-point to
represent the resultant moment, the locus of their extremities is an ellipse
when the forces receive all possible rotations about a given axis.*
[Here jLt=Y2a^^ 1 =V2a(l+zft)j8(l + ^)" 1 wbere * is the tangent of
half the angle of rotation and where i is a unit vector along the axis of
rotation, and the form of this equation establishes the theorem.]
ART. 102. The resultant of any system of forces has been
reduced in Art. 99 to a wrench which may be denoted by the
symbol (u., X) where
* (I.)
is the resultant moment with respect to the origin, where p is
the pitch, where y is the vector to any point on the axis and
where X is the resultant force. The wrench '(fyt, t\), where t is
any scalar, has by (i.) the same pitch and the same axis as (p, X).
It is therefore said to be a wrench on the same screw as (JK, X)
and it may be denoted by <(/*, X). The intensity of a wrench is
*See Joly, Trans. RJ.A., Vol. xxxii., pp. 218 et seq
164 STATICS. [CHAP. XL
the magnitude of a resultant force (TX), and the wrench (tp, t\)
has -f old the intensity of O, X).
It is often necessary to compound wrenches situated upon
different screws, and we shall investigate the simplest expression
for the wrench
(At, XH^Oup Ai) + * 2 G2> A 2 )+* S 3 , *s) ............ (II.)
which is the resultant of three wrenches of arbitrary intensity
situated upon three given screws.* Introducing a linear vector
function <f> determined by the three conditions (Ex. 9, p. 103)
we have / u = 0X if ^ = 5^! and X = 2 L X 1 ; ............ (iv.)
and thus ($X, X), in which X is arbitrary, is the general expression
for a wrench that can be compounded from wrenches on three
given screws, or conversely, that can be resolved into wrenches
on the given screws.
To reduce the problem fco its simplest form, let e be the spin-
vector of <j> and let <f> be the self -conjugate part; then
M = VeX + 0o\ = VeX - aiSiX - bjSjX - ckSk\ ......... (v.)
where a, 6 and c are the roots of <f> Q and where i, j and k
are the corresponding axes. Thus the wrench (JUL, X) may be
compounded from the wrenches (Vei+ai, i\ (Vej+bj, j),
(Ve/c+cfe, fc), situated on screws whose axes i, j and k are
mutually rectangular and which intersect at the extremity of
the vector e. The corresponding pitches are of course a, b and c ;
the latent roots of the self -conjugate part of the function <f>.
The pitch of the wrench (<f>\, X) and the vector perpendicular
on its axis are respectively (Art. 99)
-p = S0X.X" 1 , or^V^X.X- 1 ; .................. (VI.)
thus p is the reciprocal of the square of the radius of a quadric
and the vector or terminates on the surface represented by
because Vcr^HX and therefore ST=V0V^'CT(Vcr0 / cT)- 1 ; and
this surface is a quartic with three intersecting double lines
the axes of $'. (Steiner's quartic surface.)
When the origin is taken at the extremity of the vector e, the
function is self-conjugate. This point is the centre of the
three-system of screws. In terms of the pitch p and the vector ^
from the centre to any point on the axis of a screw of the
system, =XV 3 = - (VIIL)
*See Joly, Trans. JR.I.A., Vol. xxx., Part xvi., and Vol. xxxii., Part viii.
ART. 102.] COMPOSITION OF WEENCHES. 165
so that X is an axis and p the corresponding root o the function
(pp Vyp. The latent cubic of this function is (Art. 68, p. 98)
817(0 p)ij = mpm'+fPm ff --jp-, ................ (EL)
and as v\ varies, this represents a quadric surface one set of
generators consisting of the axes of screws of given pitch which
belong to the three-system. Three axes pass through an
arbitrary point, and the sum of the corresponding pitches is
constant and equal to the first invariant of $. Two axes lie in
an arbitrary plane Sa?7+l = 0; their directions (compare (vi.))
are determined by
SaX = 0, Sa^XX-Hl-O, ..................... (x.)
and the corresponding pitches are the roots of
which is the condition that the plane should touch a quadric (ix.).
In order to reduce to a canonical form the two-system of
wrenches compounded from two given wrenches ( / a 1 , X x ) and
(/x 2 , X 2 ), we assume in conformity with the foregoing a function
<j> which satisfies the relations
^X^/xp 0X 2 = /x 2 , ^>VX 1 X 2 = VVX 1 X 2 ............ (xn.)
where e is the spin- vector of <p. The function ($ Ve)p will then
be self-conjugate and will have a zero root, VX^ being the
corresponding axis, and it will be expressible in the form
bjSjp. We have (Art. 27, p. 25)
+VeVX 1 X 2 S(VX 1 X 2 )- 1 />, ......... (xm.)
and the spin- vector is deducible from the relation
2 =V{( y u 1 X 2 - / u 2 X 1 )(VX 1 X 2 )- 1 }+V6VX 1 X 2 (VX 1 X 2 )- 1 .
Operating by SVX^g we find
which gives
Taking the origin at the extremity of the vector e, a wrench
of unit intensity compounded from the two wrenches is deter-
mined by
^ = 0X=ai cos u+bj sin u=p(i cosu+j sin u) +~Vrj(i QOSU+J sinu),
X = icosu+jsinu; ................................................... (xrv.)
whence the vector equation of the cylmdroid the locus of the
central axes, and the equation for the pitch are
ri = (l> a)k sin u cos u+t(i cos u +j sin u), p = a cos 2 ^ + b sin 2 u
166 STATICS. [CHAP. XL
where u is the angle the axis makes with the vector i. The
scalar equation of the cylindroid is found on elimination of u
to be TVhfSki = (a-V)SinSji, ................... (xv.)
To show that in general a wrench may be resolved in one and
only one way into components on six given screws, or to reduce
any pair of vectors /u. and X to the forms
.................... (xvi.)
where the vectors ^ ... /JL Q and \ ... X 6 are given, we assume in
the first place
ACn=&X n , O = l, 2,3); /^=0 2 X W , (7i = 4, 5, 6); ...(xvn.)
and writing ^X x + 1^ + t s \ s = T I} 4 X 4 + 5 X 5 + V^c = T 2 .(xvin.)
we have JUL = fa^ + 2 T"2> X = r x + T 2 ;
or Tx-^-^s)- 1 ^- ^X)i T 2 = (# a -#i)- 1 (M-0 1 X) ....... (xix.)
Thus the vectors T X and r 2 are generally determinate and the
scalars t follow from (xvm).
Ex. 1. The locus of feet of perpendiculars from any point on the
generators of a cylindroid is an ellipse.
[This is evident from the form of the equation (see Ex. 7, p. 64)
Ex. 2. Find the locus of intersection of screws of the three-system
ft=<A. whose axes are coplanar with the origin.
[If jLt=<A.=^>A+V??A, fjf=<f>X.'=p f \+V'r)M the axes intersect in ?;.
Hence (<-Y?7-p)(<--V?;-jt>') destroys every vector coplanar with A and
X! and in particular it destroys 77 if BrjX.XfO. Eliminating p and p' from
(< Y^-JD)(^> Y^ p')?7=0 we have the equation of the locus which may
be written in the form
which should be compared with (vn.).]
ART. 103. To give an example of applying quaternions to a
problem in statics, consider the case of a chain lying on a smooth
surface and acted on by any force. Let be the force per unit
mass, v the normal reaction per unit length, w the mass of the
chain per unit length, and P the tension of the chain.
For equilibrium of an infinitesimal element at the extremity
of p,
0, ............ (i.)
the pull back at p being PUd/> and the pull forward at
being +PUd / >+d(PTJd / >). When the length of the chain is
ART. IDS.] CHAIN LYING ON SURFACE. 167
taken as the independent variable (Art. 85, p. 133), this may be
written
P. p "+py+f +l/ =0, S/>y = 0, Sv/>'=0; ........ (ii.)
and in virtue of the conditions it separates into
P.p"+wp'-*V P 't+v=Q, P'-S&>'=0, ......... (m.)
remembering that Tp' = l so that Sfp'- 1 ^ S//.
In certain cases the second of these equations can be integrated,
and as it may be written
dP-wSd/> = 0; P-livSgdf> = const.=P Q ......... (iv.)
is the integral in question, P being a constant.
The first equation gives the reaction
and shows that Pp ff +wp'~ l 'Vp'g is normal to the surface, or that
Ppp'+wVp'g is tangential. On elimination of the reaction (IV),
FVfTVv+wp'Sp'-^Vv^ .................. ( VL )
and the tension into the curvature into the cosine of the angle
between the osculating and tangent planes is equal to the
tangential component of the applied force per unit length which
is at right angles to the tangent to the chain.
CHAPTER XII.
FINITE DISPLACEMENTS.
ART. 104. To transfer a body from one position to another
we may commence by rotating it until lines drawn in it receive
their final directions. A translation without rotation which
brings any point into its final position will complete the trans-
ference. In quaternions* if & is the vector from a fixed point
to any point in the body, the rotation changes the vectors to
points in the body into q&q~ l , and a translation T added to this
gi^s p = T+q vq-i .............................. (l.)
for the relation between vectors or drawn to points in the initial
position of the body, and vectors p drawn from the same origin
to the same points in their final position.
This relation may be thrown into many various forms; for
example p = r > +q ^_^ q - ly r ' =T+qeq -i ............... ^
shows that if the rotation were made about the extremity of the
vector e, the successive translation must be r ; or we may first
suppose a translation ( e) effected, then the rotation about the
origin and then the translation T.
Successive displacements are compounded according to the
relations, p^T'+q'Tq'^ + q'qnq^q'- 1 .................. (ill.)
if =
and the order is all important for
p,=T+qTq- l +qq'&q'- l q- 1 .................. (IV.)
if ^ P,'=T+fBtf-\ P, = r+qp;q~ l ;
and this vector p, is not equal to p. Even the rotations are
different unless gg '= g ' g , that is unless q and q' are coplanar;
and the conditions that the order should be immaterial are
-i; gg ' = g ' g ............. (v.)
* The remarks in Art. 21 should be compared with this.
ART. IDS.] SUCCESSIVE DISPLACEMENTS. 169
Small displacements are commutative in order of application.
This is merely a particular case of a general theorem. Let any
quantity a be changed by one operation into a+f^a) where
f^a) is small, and into &+/ 2 (a) by another operation, / 2 (&) being
also small. Then to the second order of small quantities,
) ....... (vi.)
The simplest view of a displacement is as a twist about a
screw, that is a rotation about a line coupled with a proportionate
translation along the line. If r\ is the vector to any point on the
line, and PUVg the translation along the line, we have to
identify
T+g^-^+PUVg+^sr-^g'- 1 , ............. (VIL)
so that
and as it immediately appears that the first vector on the right
is at right angles to Vg, we find on resolving r along and
perpendicular to Vg,
.......... (vm.)
and of these the first is the equation of the locus of the extremity
of the vector r\, or of the axis of the screw. The ratio of P to
the angle of the rotation, or P : *2Lq, is the ratio of the pitch (p)
to a whole revolution ; and the pitch is therefore
p^JL.s^ ........................... (ix.)
* L.q UVq ^
ART. 105. Continuing to employ the same notation as in the
last article, let us suppose that q and r are functions of a variable
parameter, the time t for example, and we shall have
dp = dr + Vco(/o r)dt where d* = 2Vdgg' 1 ,
dp = dr+q(Vi&)q- l dt where idt = ZVq~ l dq .......... (i.)
To prove these relations observe that
. dq~ l
............. (n.)
remembering the expression for the differential of the reciprocal
of a quaternion. This leads at once to the first relation since
pX Ap = 2V.V>X if p is any quaternion and X any vector.
The second relation is proved in quite an analogous manner.
170 FINITE DISPLACEMENTS. [OHAP. xii.
The vector r is the vector from the fixed origin of vectors p to
the variable origin of vectors or, and its derived with respect to
the time is the velocity of that origin. The velocity of the
extremity of the vector p is compounded of this velocity together
with the velocity Vco(/o r) which is at right angles to eo and to
p_ T an d equal to the tensor of CD into the perpendicular from
the extremity of /> r on co (the two vectors co and /> T being
supposed to have a common origin). In fact the vector co
represents in magnitude and direction the angular velocity of
the body.
Using fluxional notation for the velocities, we may write
r-tfco), ...(ill.)
thus analysing the instantaneous motion of the body into a
rotation round a line coupled with a proportionate velocity of
translation along the line ; or, in Sir Robert Ball's phraseology,
we have determined the instantaneous twist-velocity about the
instantaneous screw ; the expressions
^ = ScO- a T ................ (IV.)
being the equation of the line or axis of instantaneous motion
and the pitch of the instantaneous screw. (Compare Art. 99.)
When the equation of this axis is referred to the moving
origin we may write it in the form
q~ l (yT)q = Vr 1 g~ 1 fg f +cc = ^ because a} = qiq~ l } ...(v.)
for u = 2Vqq- l = 2Vq(q- 1 q)q- l = < 2q(Vq- I q)q- l = q i q- 1 by (L).
The line ?/= Vr l q^ l rq+XL being supposed drawn in the body,
the motion of the body brings it into coincidence at the proper
instant with the instantaneous axis at the time t. Also the
rotation converts L into the angular velocity vector co at the
time t. Thus in dealing with the body itself it is convenient to
use the vectors L and or, and in considering the motion of the
body with regard to external objects, the vectors co and p are
preferably employed.
Let us no longer suppose the vector GT to be constant as in (IL).
Then if the vectors p and w are still connected by the first
equation of the last article, we shall have instead of the first
equation of the present article
fyq- 1 ; ...(vi.)
and more particularly when the vector r is constantly zero,
p = 'V(ap+qtiq- 1 , p = q(ti+'Vi'a)q-' i , if p q&q- 1 ; ...(vn.)
and still more particularly
w = qiq- 1 because < = qiq- l 9 Vcoco = 0, V = 0. ...(vm.)
ART. 105.] EELATIYE MOTION. MOVING AXES. 171
What we really do here is to compare the velocities of a point
moving arbitrarily with respect to fixed objects and with respect
to the moving body. The vector GJ represents the velocity of the
point relatively to the body, while p is its velocity relatively to
fixed objects. Sometimes a notation such as
-Afi2 = 2 6gr-i ; p = Vo> p +?^; where p = qrzq~ l ..... (ix.)
may be employed but it is not very explicit to denote the
variation ^of p arising from causes independent of the rotation;
and in this notation we may replace (vm.) by
which expresses that the rate of change of the angular velocity
is independent of the rotation. We may for example suppose
i, j and k to be fixed relatively to the vectors 57, and a=*qiq~ l ,
= qjq~\ y = qkq- 1 to be unit vectors derived from these by the
rotation. In this case if p^ax+py+yz, the derived p takes
account of the variations of a, /3 and y as well as of x, y and z,
while ^ only refers to the variations of x, y and z and not at
all to those of a, /3 and y.
These results include the whole theory of fixed and moving
axes, there being now no difficulty in writing down deriveds of
any order. For example, on differentiating (vi.) again, we have
and on substituting for p, the general formula of acceleration is
p = r + Vw(p- T ) + Va)Vft)(p-T) + ?^' 1 + 2Va)gcJ3- 1 , ...(XI.)
which may of course be expressed in terms of i.
In the case of a rigid body it is frequently convenient to
replace (in.) by the relation
p~cr+Vcop, ........................... (XII.)
where cr is the velocity of the point of the body which in-
stantaneously coincides with the fixed origin of vectors p. The
acceleration of the point at the extremity of the vector p is
P = cr+Vct)o-+Vcop+Vft)Va)p, ............... (xm.)
which follows on substitution for p in the result of differentiating
(XII.). ^
As in Art, 102, we represent the twist-velocity of the body by
the symbol (o*, a>), the fixed origin being taken as base-point, and
we may replace (iv.) of the present article by
} ....... (XIV.)
172 FINITE DISPLACEMENTS. [CHAP. xn.
Ex. 1. The instantaneous twist- velocity of a body may be reduced to a
pair of simultaneous angular velocities, /3 and /5', round two lines, by means
of the relations
o-=Ya/?+Ya73', co=/3+/3',
where a and a' are vectors to points on the lines. [Compare Art. 102.]
Ex. 2. If p is the pitch of the instantaneous screw and if <o is the angular
velocity of a rigid body, the velocity of any point in the body satisfies the
relation Spor^jp ;
and vectors drawn from a common origin to represent the simultaneous
velocities of the points of the body terminate on a common plane.
Ex. 3. The locus of points having a velocity of given magnitude is a right
circular cylinder
Tp=T(cr+Ycrp) or
coaxial with the instantaneous axis.
Ex. 4. Determine the acceleration centre of a body moving arbitrarily.
[In terms of <r and w, if the acceleration of the point at the extremity of
the vector a is instantaneously zero,
where <f>p = Yup + a> Ycop. Hence typ = tuSwp V . a>Y<oci> . p + co 3 S<op and
the third invariant is m= Ycow 2 , so that
aYa>cb 2 ==(cbSw+Y . <oYa)u>-<t> 3 Sa>) . (d-+Yaxr).]
Ex. 5. The instantaneous acceleration of a point of a rigid body moving
in any manner is a linear function of the vector to the point from the
acceleration centre, or
p=((p-a) where <p=Y<bp-f-Ya>Ya>p and d = 0.
(a) The locus of points having instantaneous accelerations of given magni-
tude is one of a system of similar and coaxial ellipsoids
concentric with the acceleration centre, whose linear dimensions are propor-
tional to the acceleration.
(b) The function <f> is independent of the velocity of translation, and a
change in that velocity merely alters the position of the acceleration centre
and of the associated ellipsoids.
Ex. 6. The locus of points for which the magnitude of the velocity is
momentarily constant is the quadric surface
S(o-hV(op)(<7+Vo>p)=0 or S{d+Yo>(p-a)}<(p-a)=0;
and the locus of points for which the direction of the velocity is momentarily
constant is the twisted cubic
^ or Y{d+Y<o(p-a)}<(p-a) = 0.
(a) The equation of the twisted cubic may also be written in the form
pYcow 2 = {(o> - ta>) S (u> to) + V . to Yo> w - o) 3 Sa>} . (<r - tv + Yoxr)
or (p-a)Y<o<b 2 =^/rd+^ 2 (a)Scbd4"a)S(od)--^a)So)d,
where t is a variable scalar.
[For the twisted cubic we have cf>p+6-+ Yo>o- = t (a- + Ya>p). Compare Ex. 4.]
ART. ior.] VARIABLE PARAMETEKS. 173
ART. 106. If the quaternion on which the rotation depends is a function
of two variable parameters, % and ^, we shall write
-du, .................. (i.)
and it must be observed that co'dw+ca.dy is not a perfect differential. To
determine the relation connecting a/ and co,, suppose or to be a constant
vector and p=qT3q~ l . Then p is a function of u and v, and
(".)
Calculating the second differentials,
or, rearranging and observing that Vo/Va^A. V(i>,Va/A.=Y . Va/u),- A., we
have, because 7 is an arbitrary vector,
a ................................ (in.)
Ol> OU
But again, by the last article and in the notation there explained,
and accordingly we may replace (in.) by this new expression
0. .............................. (T.)
The results of this article have been employed in Art. 94 in connection
with the theory of surfaces.
ART. 107. In many investigations relating to rotations for-
mulae of the type *
p = y*{iycc>Sa- x P'V'y-* ........................ (I.)
present themselves, and it may not be superfluous to make a few
remarks about their reduction. It frequently happens that
a, |8 and y form a mutually rectangular unit system, and in this
case if S = aa + b/3+cy we have
p = y*/3 2 V - aa+yt&cfv/S- yy* . bp+ypcFpy-* . cy>. ...(n.)
when we apply the general relation
a*/3 = /3a~* if Sa/3 = 0, Ta = l ................ (IIL)
In order to reduce the coefficient of 6/3 for instance, it is
generally best to start from the central term, a 2 * in this case,
and to replace it by cos irx+a sin -TTOJ, and similarly for successive
reductions. Thus we avoid introducing the sines and cosines of
the halves of the angles of rotation.
It is worth while noticing that
a -i) ............ (rv.)
*It may be advisable to refer again to Chap. IV. and its examples.
174 FINITE DISPLACEMENTS. [CHAP. xn.
is expressible in terms of the whole angle using the relation
The general relation connecting two quaternions p and q and
two scalars x and y,
(p x qp-*)y=p*qyp~ x , ......................... (v.)
will often be found useful.
Ex. 1. A planet rotates about its axis y in the period %m~ l and a
satellite describes a circular orbit round the planet in the period 2n~~ l ; show
that the motion relative to the planet is represented by
p=(y-*Sy"*y* . y-t e y mt .(y- mt Sy mt )~ nt , SSe=0,
the vectors in this expression being all fixed relatively to the planet ; and
reduce the equation to
(a) By taking the epoch when the satellite is in the plane -of the equator,
the equation may be simplified to
where r is the radius of the orbit and where Tta is the angle between the
plane of the orbit and the equator.
(&) The equation may also be written
p= ra(cos wnt sin irmt cos TTO, sin irnt cos irmt)
+r/5(cos wit cos Trmt -f cos ira sin irnt sin irmt)
+ry sin TTO, sin irnt
where a=/3y.
(c) The condition for a stationary point may be written in the form
mV yfry^fty -*/J- + n/3 a y nt ay- nt /3- a = 0,
or na + mVy ~ nt /B ~ a yft a y nt ft = 0,
and this is equivalent to
n = m cos ?ra, cos Trnt = 0.
Ex. 2. Unit vectors a, /3 and y are directed respectively to the point of
upper culmination on the celestial equator, to the east point and to the
north celestial pole, while i, j and k are directed to the south point, the east
point and the zenith. Show that the vector directed to a star may be
expressed in the forms
cr = y ~ 2 /
where TTS is the hour-angle west, iry the declination, TTW the azimuth west,
and ?ry the altitude.
(a) If irb is the latitude of the place of observation, show that
*-
and obtain the quaternion equation
and hence deduce the formulae of transformation from one set of coordinates
to the other.
ART. ior.] ASTRONOMICAL EXAMPLES. 175
Ex. 3. Assuming the effect of refraction to be K times the tangent of
the zenith distance, prove that the vector to the apparent place of a star is
cr + to where
(a) Substituting for k in terms of a and ft (Ex. 2 (a)), verify the successive
steps of the transformation
= sin irb sin Try - cos 776 cos Try cos rns + /3(sin ?r& cos TT?/ - cos irb sin TT^ cos TTZ)
y cos TT& sin ire.
(6) Show that the expression for to reduces to the form
f _ ff cos ^ s * n ^ "^ y ( s * n ff fr cos rcy "" cos ^ s * n ^ry cos ^
sin TT& sin try + cos TT& cos Try cos TTS '
where /3' and y' are unit vectors tangential respectively to the parallel of
declination and to the circle of declination.
(c) If q is the parallactic angle and the zenith distance, show that
Ex. 4. An equatorial telescope in imperfect adjustment is directed to a
star, and the circle readings are observed to be (yH-y)w and (Z+Z')TT where
y and z' are small ; if for zero circle readings the direction of the telescope
is a+ot', that of the declination axis ft + ft' and that of the polar axis y-J-y'
where a', ft and y' are small vectors perpendicular respectively to a, /5
and y, show that
and neglecting small terms of the second order obtain the relation
From this and two similar equations corresponding to the results of
setting the telescope on two other known stars, deduce the errors in the
adjustment which are represented by the small vectors a', ft and y'.
Ex. 5. The unit of length, is taken equal to the focal length of a photo-
graphic telescope in perfect adjustment so that were it not for refraction
the image of a star would remain fixed on the photographic plate. Assuming
the effect of refraction to be K times the tangent of the zenith distance,
show that the image describes on the plate a curve represented by
where ZTT is the hour angle reckoned towards the west, and where cr, y and K
are three (coplanar) unit vectors fixed relatively to the plate and directed
respectively to the star, to the north celestial pole and, when the telescope
is on the star in the meridian, to the zenith.
(a) Prove that this curve represents a conic, or a portion of a conic, and
that it is the intersection of the plane and cone
and consider the arrangement of the curves for various values of K and for
stars of different declinations.
176 FINITE DISPLACEMENTS. [CHAP. xn.
Ex. 6. The positions of stars are determined by taking transits with a
telescope movable about a fixed axis. Show that the hour-angle TTZ at the
time of transit and the declination TTIJ are connected with the reading TTU of
a circle fixed to the telescope at right angles to the fixed axis by the
quaternion equation
where 6, &', c and d are constants of the instrument, a, /3 and y having the
same signification as in Ex. 2.
(a) If B is a unit vector along the axis round which the telescope turns
the equation may be written in the form
and for an ahnucantar whose line of collimation makes a constant angle (n-a)
with the vertical and is in the meridian when ^=0, the equation is
* C os to, + /cr"/ sn ira
where irb is the latitude of the place.
Ex. 7. If Her is the unit vector towards the centre of a planet ; UCT+T
the vector towards a marking on the planet in latitude I ; y the unit vector
along the planet's axis of rotation ; a the unit vector from the planet's
centre towards the point on its equator on the meridian through the
marking ; if P is the time of rotation of the planet on its axis and s the
angular semi-diameter at the time of observation, show that
where t is the time of observation measured from some selected epoch.
(a) Denoting the vector on the right by ??, show that 77 terminates on a
fixed circle and verify that
7 cosec Z = -
where ^ ?; 3 and 973 are the values of the vector 77 at three times of
observation.
(c) Show how to deduce the time of rotation.
Ex. 8. A polar axis having a fixed direction y carries a declination axis
initially parallel to /3 on which is mounted a telescope initially parallel to a.
The vectors being all of unit length and the instrument being completely
out of adjustment so that no conditions of rectangularity are even approxi-
mately satisfied, show that when the direction of the telescope is changed
to a' by a rotation round the declination axis followed by a rotation round
the polar axis, a f
while if the rotation is first made round the polar axis and then round the
declination axis, af
and prove the equivalence of these two expressions,
(a) If u and v are the tangents of half the angles of rotation round the
polar axis and the declination axis respectively, show that the vector
equation
serves to determine both u and v.
(I) Deduce from this the scalar quadratic equation in u :
Sj8(a - a') - 2wSy/3a' - ^ 2 Sy (a - a')Sy/3 - w 2 SyVy/J(a + a') 0.
CHAPTER XIII.
STRAIN.
ART. 108. Homogeneous strain converts vectors (p) in an
unstrained body into vectors (o- = <f>p) in the manner described in
the chapter on the linear vector function (Arts. 63, 64), but the
transformation is of less generality. The order of rotation from
<pa to (f>(3 to <py must agree with that from a to /3 to y in the
case of a physical strain, for otherwise a positive volume would
be converted into a negative 'volume (Art. 24). In other words
the third invariant of the function must be positive, or the
condition m>0 ................................. ^
must be satisfied. This requires one latent root of <j> to be real
and positive, and when the roots are all real this is obviously the
case. When two of the roots are imaginary, g'+\/ Ig" and
g'~-*J \g"^ the third invariant is (g'*+g"*)g where g is the
remaining latent root ; so that here again one root is positive.
It follows from this that in every homogeneous strain one
direction at least remams unchanged, for we have
U0a = Ua if <f>a=ga, g>0 ................... (II.)
If the three latent roots are positive, three lines remain
unrotated. In the case of a pure strain three mutually
rectangular directions remain unchanged, and the function <J> is
self-conjugate with positive latent roots. The decomposition of
a linear function into a self-conjugate function preceded or
followed by a rotation has been considered in Art. 70 ; and by
selecting the square root ($$') of the function 00' which has all
its latent roots positive we decompose, without ambiguity, an
arbitrary strain into a rotation followed by a pure strain.
A sphere Tp = r is converted into an ellipsoid the strain
ellipsoid*
T-V = r or
* The results of Art. 70 show that the surface is ellipsoidal.
J.Q. M
178 STRAIN. [CHAP. xm.
and the axes of this surface are parallel to the axes of / ~ 1 0~ 1 or
of its inverse <j><j>' (not $'<f>). And the ellipsoid
T<f>p = r or Sp(f>'<j>p+i* = Q .................. ( IV -)
is converted into the sphere Ter = r. The role of the functions
<p<f>' and <t><[> is quite analogous to that of the functions of
Art. 101, p. 161, denoted by the same symbols.
ART. 109. A shear is represented by the function
0p = )-/3Sap where Sa/3 = 0, .................. (i.)
for a point in the body is displaced parallel to a fixed direction
(U/3) through a distance proportional to its distance from a plane
(Sap = 0) parallel to the fixed direction (U/3). In all cases the
displacement of a point the extremity of the vector p is </>p p.
A shear accompanied by a uniform dilatation is represented by
Sa/3 = 0, .................... (II.)
the ratio of the changed volume to the original being that of g s
to unity.
The function <j>p=gqpq- l q(3q- l Sap, Sa/3 = 0, ............. (m.)
represents a dilatation and a shear followed by a rotation, and
this function involves eight constants three in U#, one in </,
three in aT/3 and one in U/3 (because Sa/3 = 0)~ just one less
than in the general function.
Omitting the condition Sa/3 = in (ill.), the function involves
nine constants, and the function
is capable of representing the most general strain which may be
produced by shifting in a fixed direction (U/3) planes parallel to
a fixed plane (Sap = 0) by an amount ( g~ l /3Sap) proportional
to the perpendicular distance from the fixed plane ; by altering
all lines in the ratio g to unity, and by superposing a rotation.
To prove this we identify
(V.)
with Hamilton's cyclic form (Art. 77) for the general self-
conjugate ellipsoidal function so that the third invariant of <j> may
be positive or that # 2 (</-Sa)>0; ......................... (vi.)
in other words we suppose <p to be a given function, and it is
required to determine a, /3, g and q. If a 2 , 6 2 , c 2 are the latent
roots of the general self-conjugate function
0'0p = & 2 p+^ S M/> + AiSXp, .................. (VII.)
= a 2 + c 2 - 26 2 , 2TX M = a 2 - c 2
ART. no.] RESOLUTION OF STRAIN. 179
(compare Art. 77 (n.) and Ex. 2), we have on comparison with (v.)
g =b, a=-X, 6/3 =,u+i AT/3 2 , ............... (vm.)
whence substituting from (VIL) in 6T/3 = T(^ + XT/3 2 ), we find
the quadratic in T/
TX 4 T / 8 4 -2(c6 2 +c 2 )TX 2 T/3 2 +(a 2 -c 2 ) 2 = 3 ......... (EC.)
whose roots are TX 2 T/3 2 = (ac) 2 . These give
and it follows from (vi.) that we must select the negative sign.
Thus we have definitely by (vin.)
g = b, a =-X, 6/3 = M -|X- 1 (a-c) 2 ; ............ (x.)
and the rotation may be determined as in Art. 70. A second
solution is obtained by interchanging X and ju..
Ex. 1. Prove that the necessary and sufficient conditions that the
function <j> should represent a uniform dilatation and a dilatation accompany-
ing a shear, are respectively
<-<7=0, W-#) 2 =0.
[These are excellent examples of the degradation of the symbolic cubic.
Art. 66, p. 95.]
Ex. 2. If the function <j> represents a uniform dilatation and two super-
posed shears,
$ = m'.
[Assuming <p=#(l--/3'Sa')(l -/3Sa), Sa/3=Sa'/:?'=0, it is necessary to
prove that g is a root of <, and that it is equal to the cube root of m. It
may be shown that the converse is also true.]
Ex. 3. The strain produced by two successive pure strains is generally
impure.
[Two functions are commutative in order of operation only if they are
coaxial (Art. 66, Ex. 2, p. 95).]
ART. 110. Lines in the unstrained body whose lengths are
altered in a given ratio g are parallel to edges of the quadric cone
T0U P =flr, or SUp(^~ 2 )U/o = ............... (i.)
one of a concyclic system ; and by (vn.) this equation may be
replaced by
> = Z> 2 -# 2 , or sint6sin'y = (6 2 -^)(a 2 -c 2 )- 1 , ...(n.)
where 16 and v are the angles a line makes with the cyclic planes
of the function <j><p. The ratio g for any direction is the
reciprocal of the parallel radius of the quadric (compare Art.
108 (iv.)), T^ = l ............................... (m.)
180 STBAIN. [CHAP. xm.
If the inclination of the vector /3 to a remains unchanged, the
condition
SU . aft = SU . 0a0/3, or Sa/3 . T0a0/3 = Sa0'0/3 . Ta/3 . . .(IV.)
is satisfied, and the locus of the vectors ft is the quartic cone
SapBatftaBftt'tft = Sa0>/3 2 . a 2 /? 2 , ............ (v.)
which has a and Va0'0o for double edges. Substituting a + foX
for /3 in this equation, we get for the edges in the plane SX/o = 0,
which passes through a,
/8 = VX(0Va(SX-W'*)*). .................. (vi.)
after discarding the factor t\ These edges are real for all
directions of the vector X, and it easily appears that the upper
sign corresponds to SU . aft = +STJ . 0a0/3, while the lower sign
corresponds to SU. a/3= SU. (f>a<j>p on comparing the signs
of Safe and S/3(f>'<j>a. The lower sign corresponds to the case in
which the angle between <f>a and 0/3 is the supplement of that
between a and ft. The vector Va0'$a alone remains at right
angles to a, and (Art. 75 (iv.)) this vector is parallel to the
second principal axis of the section of (in.) of which a is a
principal axis.
.. If an arbitrary rotation is superposed on the strain, the cone (iv.) is the
locus of lines which together with a can be unrotated lines or axes of
q(<j>p)q~ l . The latent root corresponding to any edge (ft) is (compare (i.))
T^>u/3. To determine the rotation which must be superposed on the
strain so as to leave unrotated two vectors a and 8 satisfying the condition
(iv.) we may utilize Ex. 6, Chapter III., p. 26, and find the rotation which
converts TJ$>a and U</3 into Ua and U/2, having as in (vi.) due regard
to the indeterminate sign. It is possible to superpose a rotation on a strain
so that all the lines in a plane may be unrotated. It is only necessary to
reduce the function < to the form given in Art. 109 (iv.), and we have
(vu.)
and the lines in the plane Sap=0 (or SAp=0, compare Art. 109 (x.)) a
cyclic plane of <'< are unrotated.
ART. 111. The displacement at the extremity of the vector p
produced by the strain is
S = v-p = (<p-l)p=p(Sp-i<p p -I)+ p V p -i<j>p ) ...... (i.)
which we have resolved along and at right angles to the vector p.
When unity is a latent root of the function <j>, the displacement
is parallel to a fixed plane that of the axes of <f> complementary
to the unstrained and unrotated axis corresponding to the root
unity. (See Art. 66 (x.), p. 94.)
In general, provided the greatest and least roots of 0'0 are
greater and less than unity, it is possible by the last article
to superpose a rotation on the strain so that the resulting dis-
placement may be everywhere parallel to a fixed plane.
ART. 112.] DISPLACEMENT. 181
The quantity e = Sp- 1 (0~l)p ........................... (n.)
is called the elongation, and it is numerically equal to the
reciprocal of the square of the radius of the elongation quadric
which is parallel to the vector p. This quadric may be an
ellipsoid or a hyperboloid according to the relative magnitudes of
the roots of <p Q and unity.
The component of the displacement perpendicular to p may be
written in the form
V ?7/ o = V^ / )p- 1 . / ,=Ve/) + V0 pp- 1 .p ............ (iv.)
where e is the spin-vector of $, and (Art. 75 (iv.)) the vector
V^opp- 1 is parallel to the second principal axis of the section of
(in.) of which p is a principal axis. The magnitude of this
vector (TV/)" 1 ^ l)p) is numerically equal to the area of the
triangle formed, by lines drawn along Up and along the central
perpendicular on the corresponding tangent plane of the elonga-
tion quadric the lengths of these lines being the reciprocals of
those of the central radius and the central perpendicular.
^ * ART. 112. When the strain is not homogeneous, if the point P
^ is strained to Q, the relation between the vectors p( = OP) and
o-( = OQ) ceases to be linear, but we always have the correspond-
ing differentials linearly related, or
if <r = 0(p), ..................... (I.)
d being any function of p, and <pdp being a linear function of dp
involving the vector p in its constitution. So long then as we
confine our attention to the limits of vanishing and corresponding
regions at Q and P, so that the vector p does not vary, the
treatment of this general case is precisely the same as in the
case of homogeneous strain.
In terms of the operator V,
d<r= -SdpV <r, ........................... (n.)
so that if a is any vector which is not subject to the operation of V,
0a=-SaV.cr, and 0'a=-VSac7, ............ (in.)
as we may verify in many ways* by the results of Arts. 56
and 57 ; and in the same way it is not hard to see that we
may write
2e, ............... (IV.)
*For example a=
182 STRAIN. [CHAP. xm.
where e is the spin-vector of a. Thus for a pure strain at all
points, we must have
W<7 = 0, or c7=VP ........................ (v.)
(Art. 56) where P is a scalar function of p. (See p. 74)
./ART. 113. For small strains it is convenient to change the
notation and to consider the displacement of a point produced
by the strain rather than the relation between the vectors to the
strained and unstrained positions of the point. We write there-
fore for a homogeneous small strain
<r=p + <f>p, ....... . ...................... (i.)
replacing the function <f> of earlier articles by 1 + 0, the
function <f> being now small, or T<J>p being small in comparison
with Tp. Apart from its smallness, however, the new function
is of a more general character than the old. We may for
example have the order of rotation from <pa to 0/3 to 0y different
from that from a to /3 to y without violating the physical reality
of the strain. In fact the ratio of volumes is now
^l I m" (in
' " ''
Sa/3y
and m" is small in comparison with unity.
Small strains are superposable (cf. Art. 104 (vi.), p. 169), or
(l + 1 )(l + 2 )/> = (l + 1 + 2 V-(l + 2 )(l + 1 ) / o, ...(Hi.)
because we agree to neglect the terms of the second order 0!0 2 />
and 2 0ip-
A small strain is resolvable into a pure strain and a small
rotation by the relation
p + / o = / o + 0oP + Ve / ) = (l+V6)(l+0 )p = (l+0 )(l+Ve)p (IV.)
where is the self-conjugate part of and where e is its spin-
vector.
We may write
/0+ Ve / ) = (l + j6)/)(l + |e)- 1 = p + iep^i y o6 .......... (V.)
The strain quadric now becomes
<7 2 -2So-0 cr+r 2 = ..................... (vi.)
if p' 2 +r 2 = 0; for p = (l 0)cr if <r=(l + 0) / o, since approximately
^ non-homogeneous small strains, suppose 6(p) to be the
displacement of the extremity of the vector p. Equation (i.) then
becomes ^p + e(p), ........................... (VIL)
and for a neighbouring point
ART. us.] SMALL STRAINS. 183
Confining the attention to points in the neighbourhood of the
extremity of />, equation (vm.) is of the same form as (i.), and
the results of the present article apply if we regard the function
^ already employed as having the meaning assigned to the same
symbol in (vm.), and if we suppose that the vectors p throughout
the article are small and equivalent to the vectors dp of (vm.).
(See Art. 124, p. 211.)
Ex. 1. Interpret Hamilton's focal and cyclic transformations of a self-
conjugate function,
<f>p aaVap + b (38 ftp =gp + AS/*/) 4- /*SAp,
where <f>p represents the displacement due to a small pure strain.
[The terms may be taken separately. aaVap represents a shrinkage or an
expansion to or from one line (a) ; bpSftp represents an elongation parallel
to another. See Minchin, Treatise on Statics^ Art. 379.]
CHAPTER XIV.
DYNAMICS OF A PAETICLE.
AET. 114. The rate of change of the momentum of a particle
is equal to the applied force, or
^ t .mp=mp = .......................... (I.)
where ra is the mass ; p the velocity, mp the momentum and f
the applied force.
The moment of momentum of the particle about any point A is
V(p~a)m/j = mV( / o-a)p; .................. (u.)
and if A is a fixed point the rate of change of moment of
momentum is equal to the moment of the applied force, for
(ill.)
since "Vpp = 0. If the point A is in motion with velocity d, the
rate of change of moment of momentum is
mV(p-a)p-mVafi = V(p-a)g-m'Vdfi, ......... (iv.)
and in this case it depends on the velocity of the point A and on
that of the particle P, unless indeed the motion of A is constantly
parallel to that of P.
Since . JmT/^-roS/yf- -Stf = - js^p, ......... (v.)
the energy equation is
imT/5 8 +[slp = const =#, .................. (vi.)
and for a conservative system of forces (Art. 56 (vn.), p. 74),
(vn.)
ART. us.] DAMPED VIBEATIONS. 185
Ex. 1. If the applied force is parallel to a fixed plane SAp=0, deduce
the integral S\p=at+b ; and if it is parallel to a fixed line (ft), show that
YT . . , , ^ wnere a? 5 } a an( ^ ^ are cons t an t s of integration.
Ex. 2. If the force is directed to a fixed centre the origin of vectors p
show that
mVpp = /? = a constant vector.
Ex. 3. If T is the tangential and N the normal component of the force
and v the velocity in any orbit, prove that if is the curvature of the orbit,
v*C=lV, v=T.
[Letting accents denote deriveds of p with respect to the arc, we have
p=p'v, p=pW+p'v since v=s. Also Tp"=C and *=p'T+'Up"A r . See
Art. 117.]
Art. 115. The equation of motion of a particle of unit mass
attracted to any number of fixed centres with forces varying as
the distance is
/)* = 2a 1 (a 1 -p) = 2o 1 a 1 -p2a 1 , (l.)
the attraction to any centre being proportional to the distance
T (!""/>) an( i acting along TJ(a 1 p) towards the centre. The
scalars a v a^ etc., define the ratio of the magnitude of the
attraction of the centres to the distance, and they are positive
for attractive and negative for repulsive forces.
If a is the vector to the mean centre of the centres for the
multiples a x , a 2 , . . . a n> and if a is the sum of the multiples, the
equation takes the form
/> = a(a p), (a = Sa 1 , aa = 2a 1 a 1 ); (li.)
and the particle moves as if attracted to the mean centre.
The more general equation
p+$bfi+ep = Q, (m.)
where b and c are scalar constants, is that of the motion of a
particle acted on by a force ( cp) due to a centre at the origin
attracting or repelling (c>0 or <0) proportionally to the
distance, and also acted on by a force ( 26/3) proportional to the
velocity and accelerating or retarding according as &<0 or>0.
To integrate this equation, we assume
p = ri e^+y 2 ^+etc (iv.)
where y 1? y 2 , etc., are constant vectors and n v n z , etc., constant
scalars, and we express that the result of substituting for p in
(m.) is identically satisfied for all values of t. Equating to zero
the coefficients of e nit , etc., after substitution, we find
y(n 2 +2Zw+c) = (v.)
where y and n stand for any one of the vectors y 1 and the
corresponding scalar 7i r These conditions require all but two
186 DYNAMICS OF A PARTICLE. [CHAP. xiv.
of the vectors to vanish. The remaining two y l and y 2 are
indeterminate, and the corresponding values of n are the roots of
the coefficient of y in (v.) and are
%! =-&+#, -rc 2 =-&->, if p 2 = & a c;
^-b + tj^lq, H a =-&-^/^ : T?, if 9 2 = c-& 2 ; ...(vi.)
and the corresponding solution of the equation is
p = e~* faitf* + ?#-**) or p = e- bt (S 1 cosqt + S. 2 smqt\ ...(vn.)
the vectors y x and y 2 (or S l and <5 2 ) being arbitrary constants of
integration.
In the more general case, to solve the equation
where <^ and < 2 are two constant linear vector functions, and which
represents a damped motion of a particle such as might be supposed to take
place in a crystalline medium, an assumption of the form (iv.) gives
*i a y+w<ky + &,y = 0, ................................. (ix.)
so that the function 02+n^j+w 2 has a zero root and 7 is the corresponding
axis. The third invariant of the function must vanish if it has a zero root,
and the appropriate values of the scalars n are the roots of the equation
S(^ 8 + 6+ wS )^(*a+ w *i + w2 )f t (*2 + w ^i+ wa )'' = 0, ............ (x.)
where A, //, and v are any vectors. Solving this equation we determine six
linear functions with zero latent roots, and the corresponding axes (7^ 73, etc.),
being determined, the solution is
p-^V*. .................................... (30.)
the arbitrary constants being the tensors of the vectors 7.
Ex. 1. Show how to determine the constants of integration.
[We may have given the initial position and the initial_velocity six
constants. For example the solution of (n.) is p = a+7 a cos \fat+y% sin \lat t
and if p=/5 and p = y when = 0, we have 7i = /3-a, y^fct y.~\
ART. 116. For a force directed to a fixed centre, the origin of
vectors/), ^ U=U/>, ........................ (L)
and (Art,. 114 (in.)) we deduce at once the integral of moment
of momentum y p p=ft .............................. (n.)
where the constant /3 is double the vector area swept out by the
radius vector in unit time. Conversely if the vector moment of
momentum with respect to any fixed point is constant, that point
is a centre to which the force acting on the particle is directed,
for /3=0 = Vpp or p \\g\\ p. The orbit of the particle lies in the
P lane Sp/3 = ............................... (in.)
ART. lie.] CENTRAL CEBIT. 187
In general the vector /3 admits of transformations such as the
following (compare Art. 85 (n.), p. 132) :
where CD is the angular velocity of the radius vector, and where w
(for a plane orbit) is the angle the radius vector makes with some
prime vector or more generally where w is the scalar angular
velocity. We may also write
so that for a central force
................ (vi.)
In particular when the law of force is that of the inverse
square, the scalar m is constant, and (vi.) integrates at once
and gives
/5=-'m/3- 1 Up-fy where S/3y = by (n.) } ....... (vn.)
y being a vector constant of integration. This shows that the
hodograph of the motion is a circle whose centre is the extremity
of the vector y and whose radius is mT/3' 1 .
Moreover, substituting for p in (IL), we find the equation of
the orbit, ^/O-IT , v - ^TTT\
j8= mp Up + Vpy, ................... (viii.)
which is equivalent to the two equations
mTp = T/3 2 ~S/3 y) o, S/3 P = 0; ................. (ix.)
and which represents a conic referred to a focus as origin. If w
is the angle the radius makes with the vector y/3 we may
replace (ix.) by
Tp(l + eeoBw)=p where e = m~ 1 Tyj8, jp = m" 1 Tj8 2 , ...(x.)
and e is the excentricity andp the sewi-latus-rectum.
Taking the tensor of (VIL), utilizing (ix.) and observing that
by (x.) Ty 2 = me' 2 f)" 1 , we obtain the energy equation
where a=p(l 6 2 )" 1 is the mean distance.
Now when we resolve the velocity along and perpendicular
'/3 if r=Tpi ..... (xii.)
188 DYNAMICS OF A PAETICLE. [CHAP. xiv.
whence on substitution in (XL) we find
^ a=m ( f ? - P IN ...................... (XIIL)
\T r* aJ ^ '
which gives on integration the radius vector in terms of the
time.
Ex. 1. Deduce the usual u and 6 equations for a central orbit by
20
expressing p in the form rk v i.
[Here /a=(r+r$fc)Jfe"t, $=i*&k=hk, 6=ku*, r=r'()=-fiu' where accents
denote differentiation with respect to 6 and where u=r~\ Thus
p= -
Ex. 2. If a, ft and y are three unit vectors, a along the radius vector,
y perpendicular to the plane of the instantaneous orbit and /3 = ya ; if c is
the rate of description of angles by the radius vector in the orbit and if a
is the rate at which the plane of the orbit turns round the radius vector,
prove that the equation of motion is
[Here ^=Y^= 7 c, i=V.SQ?=ad s so that a=j8c, y=-/3d and
L a p y vpp
ft^ya-oc. Compare Art. 86. By the instantaneous orbit is meant the orbit
which a planet would describe round the sun if the disturbing forces were
suddenly removed. The equation exhibits the effect of the components of
the force along and perpendicular to the radius vector and perpendicular to
the plane of the orbit.]
Ex. 3. Express the equation of motion in a perturbed orbit in terms of
the reciprocal of the radius vector (u), the rate of description of areas (A)
and the rate (a!) at which the orbit turns round the radius vector per unit
description of angle in the orbit ; and show that it is
A V fl'u a
[We have to express everything in terms of A=TV^p=^c, of u and of a
and their differentials with respect to the angle c. Writing thus
pau" 1 we have p=Aw 2 .^-(a?*~ 1 )
Ex. 4. Express the equation of motion of a particle in the form
where u is the reciprocal of the projection of the radius vector on a fixed
plane, a is a unit vector along this projection, y is the unit normal to the
plane, /3=yo, ZTis the rate of description of the projection of areas, s is the
tangent of the angle between the radius vector and the projection, and
the independent variable is the angle in the fixed plane.
[Here p=(a+sy)u~\ a'a~ 1 =y, y'=0, /3'=-a, H<f = c if c is the angle in
the plane. The scalar equations to which the above is equivalent have been
much used in the lunar theory.]
AET. iir.] PEETUEBED ORBIT. 189
Ex. 5. Prove that the vector curvatures of an orbit and its hodograph
are
and that for a central orbit they reduce to
where ft
(a) Hence the law of nature is the only law for which the hodograph is a
circle for all initial conditions.
ART. 117. The equation of motion o a particle constrained
to move along a curve or on a surface is
P = +v ................................. (I-)
where v is the reaction arising from the constraint. If there is no
friction, the reaction is at right angles to the direction of motion
or the vector v lies in the normal plane of the constraining curve
or is the normal to the constraining surface. The condition
S*p = 0, ................................ (II.)
which is then satisfied, allows us to retain the equations (v.)
and (vi.) of Art. 114.
In terms of the deriveds with respect to the arc s of the orbit
which we now denote by p', p", etc., we have (compare Art. 85,
Ex. 1, p. 133),
p = p'v, p = p"v*+p'v, v = s, v = v'v, ............ (m.)
or in the notation of Art. 86, p. 134,
p = av, p^/Sc^+av .................... (IV.)
where v is the velocity ; and the equation of motion is
p"v*+p'v=g+ v ............................ (v.)
In the case of a constraining curve, the motion must be deter-
mined from the energy equation which is alone available for
this purpose. For a surface we have, on elimination of the
unknown tensor of y,
and in this equation v is proportional to a known function of p
the result of operating by V on the scalar equation of the
constraining surface. (Art. 54, p. 69.)
If on the other hand we seek the reaction arising from the
curve or surface, we have by (ll.)
'f, ...(vn.)
the energy equation being employed in the last transformation*
190 DYNAMICS OF A PARTICLE. [CHAP. xiv.
For a rough constraint, the equation of motion may be written in the
form
p===pV+pW==+v-flp'Tj/, SpV=0, ................ (vm.)
where n is the coefficient of friction.
Resolving along and at right angles to p' this equation gives
wf + 8p'=-nTv, v=pV-p'- 1 Yp'; ................... (ix.)
whence on elimination of TV,
pV-p'^Yp'^ -Uv.?r-i(^' + Sp') ; .................. (x.)
or again in terms of the vectors p and p, we have
7.Yp(p-^)=U(pv).Sp(p-^ ......................... (XL)
because Sp(p'~)=ftT(pv) and UYp(p-) = U(pi/). Equation (x.) or (XL)
may -be employed for a constraining surface. In the case of a' curve we
must take the tensor of each side to eliminate the unknown Ui/. We may
remark that it follows from (ix.) that if the curve is a geodesic on the
constraining surface V. vp'^Yp'^O or
Svp'=0, ....................................... (xn.)
because (Art. 90) for a geodesic p" \\v. In other words, when the direction
of the applied force is coplanar with the normal to the surface and the
tangent to the orbit, the curve is a geodesic on the surface, and in particular
this is the case when there is no applied force.
If the constraining curve or surface is in motion so that, Art. 104,
p. 168, the vector p to the particle from a fixed point is connected with
the vector & to the particle from a point moving with the constraint by the
equation
.................................. (xm.)
in which r and q are supposed to be given functions of , the equation of
motion takes the form (compare Art. 105, p. 171)
^+v, ............... (xiv.)
and for a smooth constraint,
(xv.)
being the velocity with which the particle moves along the curve or
surface of he constraint,
Ex. 1. A particle moves under gravity on a surface of revolution having
its axis vertical.
[If Jc is the unit vector directed vertically downwards, the equation of
motion is Y(p-^)v=0. Since the surface is of revolution, the vectors v,
k and p are coplanar, or Sp&v=0, so that Y&p || Y&v || Vvp. Operating on
the equation of motion by S& or Sp we find the integrable relation S/fy>p=0,
so that Sfyp=-A where h is the constant rate 6f description of area by
the projection of p on the horizontal plane. We have also Svp=0 and
=- if we write S/)=-z. From these three equations pSY&pY&j/
; and if the equation of the surface is given in the form
we may put v^p-lcfty and Vp=Yi//(;s). Hence
2 ; and by Art. 114 (vi.) or\ expressing everything in
terms of z we obtain the equation
If the surface is spherical f(z) is constant and equal to the radius of the
sphere, so that /' is zero.
ART. iir.] CONSTRAINED MOTION. 191
Again if w is the angle the plane of p and k makes with some initial plane>
from which w can be found in terms of z by the previous equation.
If, on the other hand, the equation of the surface is given in the form
Sp=/(Tyo), it may be more convenient to obtain an equation in r(=Tp)
and r by using Spp+rr=0 instead of Sfy+g=0 ; and if the equation is of
the form Sfy=/(TVfy)=/(p) we may use
Ex. 2. A particle slides under gravity within a fine smooth tube which
revolves round a vertical axis.
[The origin being taken on the axis, the vector to the particle is p=q^sq" 1
(compare p. 168), and if n is the angle through which the tube has been
rotated from some initial position,
while the equation of motion is p=gk+v where Sv^^~ 1 =0. Because the
axis of q is parallel to /, we find on elimination of the reaction v,
and in this equation n and n are given functions of t when the law of rotation
is known, and 57 is a known function of a parameter variable with the time
when the form of the tube is known. If the velocity of rotation is uniform,.
the equation integrates and
If for example the curve is a helix with its axis vertical so that
T3=a(ico$u+jsirLu) + bku we have 7 2 = -(a a + 6 2 )w 2 , and Vte 2 = a 2 , and
the equation is w 2 (a' 2 + & 2 )+?i 2 a 2 =2<7&?t-(7; and if the curve is a vertical
circle, 7=a(2*cos'W'-{-&sinw) we have
U 2 a? + h 2 a 2 cos 2 it = $ga sin u (?.]
Ex. 3. A particle under gravity traverses with uniform velocity a
smooth curve which rotates uniformly round a vertical axis. Prove that
the curve lies on a paraboloid of revolution.
[The equation of the surface on which the curve must lie is
tfTVfaP 4- 2^Sto = const.]
Ex. 4. Two particles of masses m and ra', connected by an inextensible
string which remains stretched throughout the motion, are projected from
the extremities of the vectors a and a! with the velocities ft and j8' ; prove
that the vector to the particle m during the motion is p where
the scalar n being defined by
Ex. 5. If a particle can be made by suitable initial conditions to describe
a given curve under the action of a force , show that
p' and p" being the first and second deriveds with respect to the arc and a
192 DYNAMICS OF A PARTICLE. [CHAP. xiv.
suitable constant being included in the integral which is taken along the
curve.
(a) Hence deduce M. Bonnet's theorem.
[We have m(/oV+pW)= which gives mvv'= ~S/>' and mv 2 -2jSd/>,
etc. Conversely if the condition is satisfied it follows that a particle will for
suitable initial conditions describe the curve. If x , 2 , etc., are forces under
which, acting separately, a particle can describe the curve, and if for greater
clearness we replace Js n dp by <7 n +Js n dp (the new integral being taken
from any selected point on the curve), we have
or a particle will describe the curve freely under the action of the resultant
of the forces provided its mass m and the velocity v satisfy mvP**I,m n v
initially.]
Ex. 6. Show that the condition of the last example is equivalent to the
conditions
which assert that the force must be in the osculating plane of the curve, and
that the rate of change (as we pass along the curve) of the product of the
radius of curvature into the normal component of the force is equal to double
the tangential component.
ART. 118. Tait has applied the calculus of variations in the
following manner in the determination of the curves of quickest
descent, or the brachistochrones, for a conservative system of
forces. (Quaternions, Arts. 518 and 523.)
If the integral A=(Q.Td p =\Q.dB ........................ (i.)
is taken along a curve, Q being a given scalar function of p, the
variation of the integral corresponding to a variation of the
curve is
. STdp = - Jso>V . Q . Tdp- JQSUdp . Sdp.
The symbols d and S are commutative in order of operation,
so that on integrating by parts
. dSp = [QSUd P . Sp] - sfyd (CUd/>)
where the term in square brackets corresponds to the variation
of the limits of the integral. Thus
SA = - [QSUdp . Sp] + JSo> { d(QUd/>) - VQ . Tdyo} ...... (n.)
If the integral is stationary, the variation vanishes and the
term under the sign of integration in (li.) must be zero for all
vectors Sp. And since Sp may have any direction when the
ART. us.] BRACfilSTOCHRONE. 193
curve is not restricted in any manner except at the limits, we
must have
d(QUdp)-VQ.Tdp = 0, or (Q//)-VQ = ....... (m.)
If on the other hand the curve is constrained to lie on a
surface so that Sy<5p = where v is normal to the surface, the
condition is / j \
...................... (iv.)
_ For the brachistochrone the integral A is the time of descrip-
tion of the curve or
........... (v.)
by Art. 114 (VL), so that VQ = VP.Q 3 =VP.Tp- 3 . The first
equation (in.) now becomes
d(T/j- 1 .Udp)-VP.T/j- a .d<0 or d./^+VP.T/T^^O,
or finally P+f>~ 1 - VP./> = ........................ (VL)
Tait remarks " It is very instructive to compare this equation
with that of the free path (p+VP = 0); noting how the force
VP is, as it were, reflected on the tangent of the path."
Ex. Determine the brachistochrone when gravity is the only force.
[Here VP=-/c, a constant vector, and the equation dfr l -KTp~*dt=0
shows that p" 1 a + K/(tf) where a is a constant vector which may without
loss of generality be supposed to be perpendicular to *c. Substitution gives
O, and the solution of this is
where nT. a*. Thus
/r 1 ^ -Ta(Ua+U/ctan?z,(zf-z: ))
and / 6=Ta- 1 . cos 2 n(t
and on integration
which represents a cycloid. (Tait's Quaternions, Art. 524.)]
J.Q. N
CHAPTER XV.
DYNAMICS.
ART. 119. Let m v w 2 , etc., be the masses of particles of any
dynamical system which are situated at the extremities of the
vectors p v p^ etc., drawn from a fixed origin. By Newton's
second law the equation of motion of the particle m x is
where ^ is the force external to the system which acts on m.^
and where 12 is the force due to the interaction of m 2 on m v etc.
By Newton's third law action and reaction are equal and
opposite, or
o, ................ (ii.)
these being the conditions that f 12 and 21 should equilibrate.
Hence by adding equations such as (i.) for all the particles, and
by adding the results of operating on these equations by Vp lt
V/o 2 , etc., we obtain the equations
Sm^ = 2, Sm^PiA = EV P1 , ............... (m.)
which are independent of the interactions of the particles.
Attending to (n.) the rate of change of kinetic energy of the
system of particles is evidently
and because (n.) implies 12 II ft /> 2 we see that this ^ s ^de-
pendent of the interactions provided the relative velocity of
every pair of particles is at right angles to the line joining
them or in other words, provided the distance between every
pair of particles remains unchanged.
Writing
so that M is the total mass of the system, p the vector to the
centre of mass, f the resultant external force, rj the resultant
ART. 120.] SYSTEM OF PARTICLES. 195
moment of the external forces with respect to the origin as base
point and 9 the resultant moment of momentum with respect to
the origin, the equations (in.) become
Mp=& 0=>? ............................ (vi.)
When the external forces are zero, and q vanish and the
integrals of (vi.) are
Mp = at+P, = y, ...................... (VII.)
where a, /3 and y are constant vectors ; and when the internal
forces are given functions of the distances between the particles,
we have also in this case the integral of energy
ART. 120. With reference to a point moving in any arbitrary
manner, the extremity of the vector e, the moment of momentum is
e / >-66); ....... (i.)
and (vi.), Art. 119, may be replaced by
Mp=i 0. = *-#V(p-e)e, .................. (II.)
where ^ = 77 Ve is the resultant moment of the forces about the
extremity of e. In particular when e terminates at the centre of
mass, the equations are
where and ^ refer to the centre of mass. These equations are
of the same form as those of the last article. We may note that
in general
e^e-MVpfi^e.- MVp.fr, .................. (iv.)
where p e = p e.
Ex. 1. Find the locus of points fixed in space about which at any instant
the moment of momentum is a minimum.
[If the extremity of terminates at a fixed point 6e=0-MV^ and the
locus of points for which T0 has a given value is the right circular cylinder
T(0- J/Vp)=T&. Writing 0=M(pp+'V^p) we have
The locus is the line MVp=VQp . fr\ Compare Art. 99, p. 156.]
Ex. 2. A point moves in such a manner that the moment of momentum
with respect to it is constant. Determine the particulars of the motion.
[If 6e is constant, the relation (iv.) lfVp f p e =-d+0 e +MVpp gives, on
differentiating twice and utilizing the equations of motion (Art. 119 (vi.)),
because e is constant. Forming the vectors of the products of right and of
left hand members of the first and second of these three relations, and
also forming the scalar of the product of corresponding members of the
three relations, we obtain the equation
196 DYNAMICS. [CHAP, xv,
so that e is expressed in terms of quantities which are known when tht
motion of the system is given. There are thus two paths corresponding tc
the double sign symmetrically placed with respect to the path of the centre
of mass.]
Ex. 3. Refer the equations of motion to variable axes.
[See Art. 105 and the formulae of differentiation (vi.) and (XL), p. 170.]
ART. 121. In the case of a rigid body, let e be the vector tc
any point fixed in it and let a> be the angular velocity. Then bj
Art. 105, p. 170, > \r t \ <v
r -= -
because the velocity of the point in the body at the extremity oJ
p l relatively to that at the extremity of e is due to the angulai
velocity w. Equation (i.) of the last article may now be replacec
so that 6 e is a linear function of w. The linear function <j> iis
fixed relatively to the body because the vectors p l e, etc., an
fixed in the body, but in considering the rate of change of <p e a
we must take account of the change of orientation of the body a
well as of the change of . We have (Art. 105 (ix.)),
d 3(0 eo)
and equations (n.) of the last article become
e)'6; ......... (iv.
and when e terminates at the centre of mass (Art. 120 (ill.)),
Mp=g, 0t+Va>0ft> = j7 ,. : ................... (V.
if (Art. 120 (IV.)) ^ = ^co-AfV( y o~e)Va)(/3-e) refers to th<
centre of mass.
If the body has a fixed point, the extremity of e, (iv.) reduces tc
VI.
In general the vector <p& is the moment of momentum of th<
body with reference to the fixed point which instantaneouslj
coincides with the extremity of the vector e, and the moment 0"
inertia round any line (Uco) through that point is
co.^-e) 2 , ............... (VII.
and this is numerically equal to the reciprocal of the square o:
the parallel radius of the quadric
=-l ......................... (vm.
The function e may be called the inertia function corre
spending to the extremity of e.
ART. 121.] EIGID BODY. 197
^ The principal axes of the body through the point are the axes
of the self-conjugate function e , and the moment of inertia
round a principal axis is maximum, or minimum, or at least
stationary in value. If the extremity of e is fixed in space as
well as in the body, so that the body moves about a fixed point,
it appears from (vi.) that when the body is set rotating under
no forces about one of these principal axes, it will rotate
permanently round it. For we have Vo)0 e &) = if aj is along a
principal axis, and e w = by (vi.); hence w = since the function
has not in general a zero root.
The energy equation (Art. 119 (iv.)) easily reduces in terms
of e and w to
} = -Se-S7 e , ...(IX.)
where ??e = 2V(/) 1 e)^; and when e terminates at the centre of
mass j
(X.)
Ex. 1. Prove the relation (in.) by direct differentiation of the explicit
form <
[Wehave
=Y.Yco (pj-e) Vo)(p 1
by (L). The first terra on the right vanishes. The third is
V<o(p 1 -)Sa)(p 1 -6) or V.coVfa^Vcjfo-e).]
Ex. 2. If J is a principal moment of inertia at the extremity of the
vector e, or in other words a latent root of < e , show that
P - 2?i"/ 2 + (ri /2 + ?0 /- (n "ri - ) = 0,
where ?i", n' and n are three positive scalars, namely,
[See Elements, Art. 417, and observe that <^ e a)=^a)-
Compare Art. 65, Ex. 1, p. 92.]
Ex. 3. The function <* corresponding to the extremity of the vector
drawn from the centre of mass is
where < corresponds to the centre of mass; the principal axes at the
extremity of 7 are the normals to the three conf ocals.
which pass through that point ; and the locus of points at which one of the
moments of inertia is equal to I is the quartic surface
[If ^a=/tt=<j[>a + JfYwYaZ7 5 we have
etc., and u=H~ l I- Tor 2 . Compare Art. 101 (xx.) and (xxi.), p. 162.]
198 DYNAMICS. [CHAP. xv.
Ex. 4. A body under no applied forces moves about a fixed point.
The equation of motion <0^V<i>9co==0, furnishes the integrals
where and h are constants of integration. Interpret this result.
(a) The equation Seo<co= -A 2 may be regarded as representing an ellipsoid
fixed in the body which rolls upon a plane fixed in space, and represented by
the equation So>0=-A 2 , the point of contact being the extremity of the
vector o).
(6) The equation ^ 2 Soxjf>co-A 2 <jf)a) 2 =0 represents a cone fixed in the body
which is the body locus of the instantaneous axis of rotation ; and because
the rate of change of <o is the same with respect to the body as with respect
to lines of reference fixed in space (Art. 105 (x.)) it follows that this cone
rolls on the space locus of the instantaneous axis.
(c) The extremity of the vector co describes in the body part of the curve
of intersection of two quadrics fixed in the body (the polhode) Sto<w = -A 2
and So)(jS 2 o) = s , and the locus of the same point in space is a plane curve
(the herpolkode).
(d) The vector 0, though fixed in space, describes in the body the cone
g^BQ^B- A 2 2 =0 where #=T0 is the constant tensor of 0, and the extremity
of traces out part of the sphero-conic in which this cone cuts the reciprocal
(e) The reciprocal quadric, fixed in the body, passes through a fixed point
in space, and the central perpendicular on the tangent plane at this point
varies inversely as the angular velocity.
(/) The relations
S<o> (io + VCDO>) = 0, S(u> (oi - Vaxo) = 0,
in which a> is the rate of change of u> with respect to the body, may be
obtained by differentiating the equation of motion. Hence
<f><D . (So wu> + Vcocb 2 ) = A 2 Vw (iti + "Vox!))
and <w (SoxJuoi + Vw w 2 ) = 7i 2 Vo>YtJb (ui + V(ou)) ;
and the vectors w, o> and w satisfy a condition
SVw (to - VOKU) Vtb (w + Vcoco) = 0,
which is independent of the constants of the body. The corresponding
relation SV w (D, 2 w - 2Vo>D,o>) VD^D^CO =0
connects o> and its first and second deriveds DO> and D c 8 o) with respect to
fixed axes.
(g) Knowing <u at any instant and its first and second deriveds with
reference either to axes fixed in the body or in space, the function < is
determinate to a factor.
Ex. 5. The angular velocity of a body moving under no forces about a
fixed point is expressible in terms of elliptic functions by the relation
where x^J^xP-Ix-J and
a being a constant imaginary vector, ^ being a linear function coaxial with <f>
and having for its latent cubic 4# 3 Ig J 0.
[Compare Art. 84, p. 124. Here the assumed expression for cu gives
or (<j + #)~a + (fa + xyVafam^x) = 0,
ART. 121.] RIGID BODY. 199
where m-^x) is the third invariant of fa+x. We may obviously take the
first invariant of fa to be zero without loss of generality, so that the latent
cubic of the specified type, and the differential equation for % is reduced to
Weierstrass's standard form.
The function fa is of the form fa=a+bx+c\jr where y an d ^ are the
auxiliary functions for fa and when the first invariant is taken to be zero,
3a+2bm"+cm'=Q. The scalars b and c are arbitrary constants of integra-
tion. Assuming aui+vj+wk where i. j, Ic are the axes of fa we see that
A^t l =(B-C)vw, Bv=(C-A)iou, Cw=(A-B)uv, A, B and C being the latent
roots of < the principal moments of inertia. Thus
AB
and the latent roots of fa are $b (B +C- 2A) -%c(CA+AB-%BC\ etc.
Moreover since by definition of a, we have Sa<a = 0, Sa< 2 a=0 and also
a 2 = + l as maybe easily shown, we find Sii>cl>ij)=Sa<l>(fa+x)a=cABC and
(<hu)' 2 ==Sad> 2 (fa+)a==--bABC) or in the notation of the last example.
cAJBC=-tf*n.dbABC=&*.]
Ex. 6. Resolve the vector of angular momentum <o>, along and at right
angles to co, and investigate the relation of the components to the quadric
-
[Compare Art. Ill, p. 181.]
Ex. 7. The motion of a freely moving body is known, and it is required
to determine as far as possible its dynamical constants.
[The mass cannot be determined, but if we know the particulars of the
motion of three points, the extremities of j, e 2 and 3 , we can find o> from the
two equations ^ - e 2 = Vco (e, 2 ), ^ e 3 = vo> ( 1 e 3 ).
In the next place, to find p, the vector to the centre of mass, we have
e-), and fe
and because p=0 the second of these relations gives p on solution of a linear
equation.- To find the function < corresponding to the centre of mass, we
differentiate CD twice and use the results of Ex. 4, (/) and
Ex. 8. Given four particles whose united mass is that of a given rigid
body, it is required to connect the particles by a light frame- work, so that
the 'dynamical constants of the system may be identical with those of the
body.
[If <A.= SwjpjSpjA. where A. is an arbitrary vector, and where the
vectors p l9 etc., are drawn from the centre of mass of the body to terminate
at the particles of mass m 1? etc., the problem is solved when we reduce the
function <j> to the form
, where
a, 5, c and d being the masses of the four particles and a, ft, y and 8 being
their vectors of position. Now for some scalar #, we have
and we also have (Art. 65, Ex. 1, p. 92)
The second of these serves to determine #, for it reduces to m=z3
Substituting a for X in the first, we find ^a=xbcdV^(ft - 8)(y - 8) ; and when
we operate with Sa, S/3 and Sy on this and similar expressions we have
8a^a=--aPbcd(b + c+ch 9 etc., Sa^/3 = etc.=AW. It easily appears that
the six relations in a, p and y imply the remaining six involving 8 when
200 DYNAMICS. [CHAP, xv,
Assuming first any vector a which satisfies the condition
a = - a?bcd(b + c+d)
that is any vector which terminates on a certain quadric we have next the
two relations $a\ls/3=a?abcd, S/3^/3=-a?acd(a+c+d) which require the
vector ft to terminate on a conic. Selecting B there remain the three
equations Sa^y = B/3^y=^abcd y Sy^-y -a?abd(a+b + d) which determine
y as the vector to a point of intersection of a line and a quadric. Finally,
we have S= -d-^aa+bft+cy).]
ART. 122. When an impulse acts on a system of particles, the
velocity of the particle m l is changed from /5 LO to fa where
where X x is the external impulse acting on m l and where X 12 , X 13 ,
etc., are the impulsive actions of the particles m 2 , ra s , etc., on m r
These impulsive interactions satisfy conditions analogous to (n.)
of Art. 119, and we obtain the equations
2m 1 ( / 5 1 - / 5 LO ) = EX 1 , 2m 1 Vp 1 05 1 -A,o) = SV / ^X 1 , ...... (n.)
which are independent of the interactions. The work done on
the particle m 1 by the impulse is (Thomson and Tait, Art. 308)
-iS(A+/Ufti+Xu+Xis+etcO, ............... (in.)
and the total work done on the whole system is
For a rigid body it is frequently convenient to define the motion
by the velocity (cr) of the point of the body coinciding with
the origin and the angular velocity (o>). Thus /d 1 = <r V/)^,
and if x = 2Xij ^^V PI \ V ^SV/^Vo^, ............ (v.)
so that X is the resultant force and ju. the resultant moment of
the impulse with respect to the origin while <j> is the inertia
function corresponding to the origin, the equations (n.) become
Jf(<r-Oo-Vp(o> o> )) = X, -WVp((r <r ) + 0(a> o) )5=^t; (vi.)
and because X 12 is parallel to the line joining two particles and
therefore perpendicular to fa fa and to pi,Q-p. 2>Q , the expression
for the work done is independent of X 12 , etc., and reduces to
W^-iSCcr+o-oJX-iS^ + ao)/*, ........... :...(vil.)
because we have
When the origin is taken at the centre of mass, (vi.) becomes
Mfa-o-Q^X, ^ofa ^o)"/*' ............... ( VIIL )
where ^ refers to the centre of mass, and thus we have at once
; ............... (IX.)
ART. 122.] IMPULSES. 201
or, in the language of the theory of screws, when a free body
having an instantaneous twist velocity (CT O , o> ) is acted on by
an impulsive wrench (p., X), the instantaneous twist velocity
immediately after the impulse is (o- +^- 1 X, ft> +0 ~V)> the
centre of mass being the base-point. (See p. 171.)
When the origin is taken at an arbitrary point, we may replace
(vi.) by
Jf(o (70-7/3(0) -eo )) = X, &(-b) = M-VpX, ...... (X.)
where (<r, o>), (<r , co ) and (/*, X) are referred to the origin as
base-point and where <f> corresponds to the centre of mass. This
is easily shown in various ways.
The form of the expression (vn.) is independent of the choice
of base-point. In particular when the base-point is at the centre
of mass, we find from (VIL), (VIIL) and (ix.),
W = -
(XL)
Ex. 1. Prove that the solution of (vi.) is
(w - o> ) (m + MBp^xp + M 2 Sp<f>p . fr)
; M(o cr ) =
where m is the third invariant of <j> and where x an d ty are the auxiliary
functions.
[Compare Ex. 5, Chap. VIIL, p. 102.]
Ex. 2. A rigid body is moving in any manner and an impulsive force is
applied to a given point of the Body so as to cause that point to move
instantaneously with a given velocity. Determine all particulars.
[The centre of mass being taken as base-point, and a being the vector to
the point in question and d being the velocity of the point, the equations
M(<r cr )=A, <(co-a) )=VaA, <r Vaaj = d
serve to determine the unknowns o-, a> and A. We have on elimination of
o- and A, <#>a)-J/'ayaco=:^o) +J/Va(d~o- ) ; and by Ex. 5, Chap. VIIL, the
solution may be written
(o> - o> ) (m - MSacf> X a + J/ 2 a 2 Sa<a) = J^Va (a - d ) - J/ 2 V <aVaYa (a - OQ),
where d =cr Yaw is the initial velocity of the point. Hence in terms of
tu - oj as given by this equation
o--o- =d-d +Ya((o-o) ) and A^J/" 1 ^-^).]
Ex. 3. A rigid body is moving in any manner. Suddenly a line in the
body is constrained to move in a definite manner.
[If a and /3 are the vectors from the centre of mass to two points on the
line, we may suppose the impulsive wrench to consist of forces A and A'
applied at the extremities of a and /3. Hence
where d and ft are the velocities of the extremities of a and ft. From the
first and second equation we deduce S(^-a)<5f>(o)-w ) + i/ f Sa/?((r-cr )=0,
which asserts that the moment of momentum about the line is unchanged.
202 DYNAMICS. [CHAP. xv.
We also have a>=(d-/3+#)(a-/3)~~ 1 , where x is a scalar to be determined
by substituting for o> and cr in the equation just found. Solving the linear
equation for x we find CD, and hence <r and A and A'.]
Ex. 4. A rigid body is moving in any manner. It is required with the
least possible expenditure of energy to cause a given point to move in a
given manner.
[Writing equation (xi.) in the form
TF- - %(M(a + Vao)) 2 + SOM/KO) + J( M(<^ + Vaa> ) 3
we express that this function of w is a minimum. We find
<jxo JS/dYao) = M Yad,
and as in Ex. 2, this gives
co (m - J/Scc<xa + J/ 2 a 2 Sa<a) = J/
and substituting in cr=d+Yaco, in <(a)-w )=/* and in M(cr - o- ) = X, we
determine the impulsive wrench and the instantaneous twist velocity.]
Ex. 5. If p and p' are the pitches of the screws of an impulsive wrench
and of the instantaneous twist velocity produced by the wrench on a free
quiescent rigid body ; if also 13 and T$ are the vector perpendiculars from
the centres of mass on the axes of these screws, lfa-=lf(2/-|-5J')w = A,
^ = (p + ztr) A = <a>.
(a) Hence in terms of A and o>,
r 1 , CT'= Jf-'YAor 1 ;
(6) The shortest vector from the axis of the impulsive screw to that of
the instantaneous screw is ^(STiysT'" 1 - 1).
(<") Show that
and express the moment of inertia about the line through the centre of mass
parallel to the instantaneous axis in terms of jp, p'^ 7 and tu'.
(d) The cosine of the angle between the axes of the two screws is
; and if the axes are parallel, that of the instantaneous screw
passes through the centre of mass or else the instantaneous motion is a
translation. In the former case the pitch and vector perpendicular on the
axis of the impulsive screw satisfy the condition
Ex. 6. Determine the dynamical constants of a free body by observing
the effects produced by impulsive wrenches in starting the body from a
given position.
[If p is the vector from a fixed origin to the unknown centre of mass,
if an impulsive wrench is (/x, A) and the corresponding twist velocity is (a, <o)
for the fixed origin as base-point, the equations are (compare (x.))
together with others with accented letters o-', co', //, A/, cr", a>", //', A" for
other impulsive wrenches and the corresponding twist velocities. From
these equations J/", p and <J> (corresponding to the centre of mass) are to be
determined. The mass follows at once from the first equation, and we have
M= S Aw (So-co)- 1 = S A V (So-V)- 1 = S A
ART. 122.] THEORY OF SCREWS. 203
The vector p is given by
V (o M~ l X) (cr 7 - M~ l X') = W/xoV/W = - pS (<r - if- 1 A) '.
And the function <p can be found from three couple equations. Some rather
elegant identities connecting the wrenches and the twist velocities may be
deduced from this beautiful problem of Sir Robert Ball's.]
Ex. 7. An impulsive wrench of given pitch and intensity is applied to
a free quiescent rigid body. The axis of the screw of the wrench passes
through a fixed point ; find the direction of the axis so that (a) the kinetic
energy, or (6) the angular velocity, generated by the impulse may be as
great as possible.
[The base-point being taken at the centre of mass, we have Jf.cr=A,
y)A where TA, p and y are given. The kinetic energy is
---|jl/~- 1 A 2 , and if this is a maximum subject to
the condition that TA is given, we have(p9~Vy)^>- 1 (p-hV7)A=^A where
g is a scalar a latent root of the self -con jugate function on the left, and for
a maximum g is the greatest latent root. The kinetic energy is
The least latent root answers to minimum kinetic energy. For a maximum
or minimum angular velocity deal similarly with the equation
Ex. 8. An impulsive wrench (^ A.) is applied to a free rigid body
moving with the instantaneous twist-velocity (cr, w). The change in the
kinetic energy is T _
where T is the kinetic energy that would have been generated were the
body at rest.
(a) With the same meaning for T, show that the wrench
on the arbitrary screw (/i, A) leaves the kinetic energy of the body
unchanged.
(6) The centre of mass being base-point, any wrench on the screw
(<o>, Mcr\ acting on the body when moving with the twist- velocity (tr, w),
leaves the screw of the instantaneous twist-velocity unchanged.
Ex. 9. Two bodies collide. Assuming that the impulsive interaction up
to a certain stage of the impact is equivalent to a single force (A) at the
point of contact, the equations of motion are
M l (oV - crO = A, <k - o>i) = Va x A ; J/^oy - 0-2) = - *i 4>i (^ ~ w a) = - Ya 2^
where (o^, a^) and (o-/, to/) are the twist-velocities of the^body M just
before the commencement of the impact and at the particular^ stage of
the impact under consideration, the centre of mass of M^ being base-
point ; where <fo is the inertia-function of M l corresponding to its centre of
mass, and where 04 is the vector from the same origin to the point of
contact ; o- 2 , o> 2 , <r 2 ', o>./, < 2 and a 2 being in like manner related to the body
J/2 and to its centre of mass.
(a) The relative velocity of the points of the bodies in contact is
^ + Vw/oj - o-./ - Vco 3 'a 2 = o-! + Vw^ - o- 2 - Va> 2 a 3 + ( J/T 1 + ^ 2 ~ 1 ) *
+ V . <r 1Va i A. . % + V . <&f * V og A . a 2 ;
or briefly, it is r = r + * A,
204 DYNAMICS. [CHAP. xv.
where < is a certain self -con jugate function determined by the circumstances
of the impact and where r is the initial relative velocity of the points in
contact.
(b) For perfectly smooth bodies, VAv = 0, where v is the normal to the
bodies at the point of contact, and the value of A, corresponding to the end
of the "first period" of impact is
and the twist- velocity of the body JI/i immediately after the impact is
where e is the coefficient of restitution.
(c) The total loss of kinetic energy is
(d) For perfectly rough bodies, VrV=0. The value of A, corresponding
to the end of the first period of impact is X= $~ 1 r, and the twist- velocity
immediately after the impact is
(e) For perfectly rough bodies, the loss of kinetic energy is
ART. 123. When a rigid body is not perfectly free but
constrained in any manner an impulsive wrench will in general
be partially neutralized by the reaction of the constraints.
Referred to the centre of mass as base-point, we have for a
quiescent body,
X X v , ^>co = //, /oc,, ..................... (i.)
where (JJL, X) is the impulsive wrench and (JUL,, X,) the wrench on
the constraints, or where ( ^, X,) is the reaction of the
constraints. In order to determine the instantaneous motion
produced by the impulsive wrench (p, X), it is necessary to know
the evoked wrench (^ X,). We consider the case in which the
constraints are smooth, or so that no evoked wrench can generate
any motion. In this case the work done by the wrench (^ X,)
must be zero, or we must have (Art. 122 (vn.))
SGv + X/r) = 0, ........................... (II.)
where (yu,, X,) is any wrench arising from the constraints and
where (<r, w) is any possible twist velocity of the body. The
screws of (JUL,, \) and of (cr, co) are said to be reciprocal when
this condition is satisfied; and for smooth constraints, every
possible twist velocity is reciprocal to every possible wrench
arising from the constraints.
A body with one degree of freedom can move only one way
from a given position, by a twist about some definite screw
(i ^i)- -^ kdy with two degrees of freedom can move in a
singly infinite variety of ways from a given position ; if (<r x , o^)
and (<r 2 , co 2 ) are two screws about which the body can begin to
ART. 123.] CONSTRAINED BODY. 205
twist, it can begin to twist about every screw of the two- system
( X i'i+ x 2'& ^i^i+^V*^)' where x l and #<> are scalars, as easily
appears from the composition of small displacements (o-jd^, wjd^)
and (<7 2 d 2 , co 2 d^). Similarly a body with n degrees of freedom
can begin to twist about any screw of the Ti-system (S^cr^ 'Zx^),
where (cr 1? o^) . . . (0^, <*> n ) ^^e n independent screws about which
the body can, begin to twist ; and being given ?i independent
screws about which the body can begin to twist, all possible
initial motions belong to a given system of twists. Every
wrench reciprocal to n independent screws of the freedom is a
wrench arising from the constraints, for every such wrench is
reciprocal to every possible twist on account of the linear
character of the condition of reciprocity (n.), and no such
wrench can generate any motion in the body. By expressing
that a wrench (/*,, \) is reciprocal to n screws of the freedom,
the number of its arbitrary constants is reduced from 6 to 6 ^
since n conditions (n.) must be satisfied ; and thus the screws of
the constraint compose a system of order (6 ri). This system
can be determined when the system of the freedom is known,
and conversely.
Again knowing the system of screws of the freedom we can
determine what Sir Robert Ball calls the screws of the reduced
wrenches. A reduced wrench causes no reaction on the con-
straints ; it produces the same initial motion as if the body were
perfectly free. In equations (i.) the wrench (JU.JJL,, X X y ) is a
reduced wrench, or (<&>, M<j) is the reduced wrench corresponding
to the twist velocity (a-, a?). The system of screws of the reduced
wrenches is (tj&x^ H Sa^oi) when that of the freedom is
Suppose now that we select n independent screws of the
'ft-system of the reduced wrenches and 6^ screws of the
((j_ 7&)-system of the constraints, and that (Art. 102) we resolve
an impulsive wrench (/oc, X) into its components on these six
screws, we shall have (compare (xvi.), p. 166),
X = X' + \, ..................... (HI.)
where (/, X') is the component of (p, X) belonging to the system
of the reduced wrenches and where (^, X,) is the component
belonging to the system of the wrenches of the constraint.
The instantaneous twist velocity is then given by the relations
1. Prove that
,, ?
represent respectively a three-system of screws (/x, A) and the reciprocal
three-system (//, A/), <f> being a given linear vector function and A, and X'
being arbitrary vectors.
206 DYNAMICS. [CHAP. xv.
[Compare Art. 102 (iv.), and observe that if (//, A') is reciprocal to every
screw of the system (p, A), we must have SOuA'+ft'A) = or SA(<T+///)=0
for all vectors A.]
Ex. 2. Determine triads of co-reciprocal screws of a three-system.
[If the screws (^ A^, (/i 2 , A 2 ) and (u 3 , A 3 ) of the system //,=< A are
mutually reciprocal, S A^ + <') Ag = 0, SA 2 (<+ <j>')A 3 =0, SAj^+^A^O ;
or A 15 A 2 and A 3 are parallel to mutually conjugate radii of the quadric
const. Thus
where ?', ./ and k are three mutually perpendicular unit vectors.]
Ex. 3. Determine sextets of co-reciprocal screws.
[Take any triad of co-reciprocal screws of a three-system //, = cf> A, and any
triad of co-reciprocal screws of the reciprocal system /A= - <'A.]
Ex. 4. Resolve a wrench (or twist) into its components on six co-
reciprocal screws.
[If (/AU Ai)...^, A 6 ) are the six co-reciprocals, we can find a linear
function <f> so that (j^, Aj), (jt% A 2 ) and (ft 3 , A 3 ) belong to the system ///=<A';
and then (ft 4 , A 4 ), (/* 5 , A 5 ) and (/^ A G ) will belong to the system //'= -<'A".
"
4 , 4 , , .
We assume for the given wrench (ft, A) that ju,=<A'-c'A" and A=A'+A" ;
whence we have generally A' = (< 4- $')~ V + ^) and A" =-(< + ^O" 1 ^ - </>^),
and it only remains to resolve A r along A t , A 2 and A 3 , and A" along A 4 , A 6 and
A 6 in order to obtain the required relations /A =2^^, A=2,^ 1 A 1 .]
Ex. 5. Find the (n - 6)-sy stern reciprocal to a given ^-system.
[This has been effected in Ex. 1 for 9&=3. Let ft =4, and for any three
screws of the system construct the function $. Resolve any fourth screw
(/*, A n ) as in the last example, so that ju, n =<A'--<'A" and A W =A / +A", and
take two vectors A 5 and A 6 which with A" compose a mutually conjugate
triad with respect to S/o (< + <$>') p= const. Then
is the two-system reciprocal to the four-system. To determine the four-
system reciprocal to a given two- system, take any function <j> satisfying
y^ <A 1? /A 2 =<#>A 2 , where (p, 19 A x ) and (^ A 2 ) are two screws of the two-system,
and determine the vector A 3 conjugate to A x and A 3 with respect to the
quadric Sp(<j>+cl>')p=cQTLst. The required four-system is
where A' and # 3 are arbitrary. Similarly we may proceed in other cases.]
Ex. 6. Show that
S (fiX + /A) = (p +p') SAV + S(y - /) AA',
where p and p' are the pitches of the screws (/*, A) and (fjf, A'), and where
y and y' are the vectors to points on their axes. Interpret this result,
and show that
- S (ftA' + /A' A) = (p +p f ) cos u + d sin ^,
where u is the angle and where d is the shortest distance between the axes.
Ex. 7. A body twisting along the screw (o^, o) x ) is suddenly constrained
to twist along another screw (cr 2 , <t> 2 ). Determine the motion.
[If (#o- 1} tfwj) is the twist velocity just before the change and (yo- 2 ,
that just after, we have
A,
ART. 123.] CONSTRAINED BODY. 207
where (/x, A,) is the wrench arising from the constraint which produces the
change of motion. This wrench is reciprocal to (cr 2 , o> 2 ), so that
Substituting we easily find y to be given in terms of x by the relation
Ex. 8. A body oscillates under the action of a conservative system of
forces, and at a certain part of its swing the motion is suddenly changed from
a twist about one given screw (o^, o^) to a twist about another (cr 2 , o> 2 ).
Show that the twist velocities just before the sudden changes of motion at
the beginning and end of a complete oscillation are in the ratio
^o)!) ( J/o- 2 2 + S<
the base-point being coincident with the* position of the centre of mass at
the instant of the change of motion.
[This is the general case of a self-closing gate. By the last example
y(M<r} + So) 2 <o> 2 ) =*#( J/So-jO-2 4- So^cog)
and y (M 80^0-3 + So^o^) = x' (Nvf + Sw^Wi),
where x : so' is the ratio of the twist velocities just before the change from
the screw (o- 1: o^) to the screw (o- 2 , o> 2 ) and just after the change from (cr 2 , o> 2 )
back to (cr 1} coj). The system of forces being conservative, the magnitude of
the twist velocity throughout the partial oscillation during the continuous
part of the swing depends solely on the position of the body, and is the same
just after the sudden change from (o- 2 , o) 2 ) to (o^, Wj) as just before the next
sudden change from (o-j, o^) to (o- 2 , fc> 2 ). To show that x is greater than x' or
that (l/b-^H- SCD^O),) (J/^ + Swa^Tco^-^Sor^+Saj^cojj) 2 is positive, turns
on the fact that a?/3 2 +y 2 b 2 2Sa/3Sy8 is positive when a, /?, y and 8 are real
vectors. The value of this expression lies between the limits (Ta# Ty 8) 2 .]
Ex. 9. An impulsive wrench reciprocal to the instantaneous twist
velocity of a free body at the moment of its application increases the kinetic
energy.
[The change of kinetic energy (Art. 122 (xi.) is -S^<~ 1 a--pf~ 1 A. 2 , and
this is equal to the kinetic energy which the wrench would generate were
the body at rest.]
Ex. 10. Determine the dynamical constants and the constraints of a
rigid body by observing the effects of impulsive wrenches applied to the
body when placed in a given position.
[Let (jutj, AJ), (PM \i) and (c^, c^) represent an impulsive wrench, the
corresponding opposing wrench arising from the constraints and the twist
velocity produced. We know (o-j, coj) by observation that is, a screw of the
freedom. A second impulsive wrench (/A>, A 2 ) being applied, we find a
second screw of the freedom (cr 2 , o) 2 ), provided we have not o- 2 = to-^ o> 2 = t^ In
this second case, however, we have a screw of the constraint, for the impulsive
wrench (i^-t^ X^-t^) generates no motion. Administering a third
wrench we obtain similarly either a new screw of the freedom or a new screw
of the constraint ; and from the results of applying six independent wrenches,
the screw systems of the freedom and of the constraint become completely
known. These systems being known, we can by (in.) resolve an impulsive
wrench (j^, X a ) into the reduced wrench (/z/, X/) and the evoked wrench
(u, X yl ); and we have as many sets of equations Mfa- Y/)o> 1 ) = A 1 ',
^o)! /^'-VpA/ as degrees of freedom. Tor one degree of freedom, the
first equation gives the mass MB^ A/ : Sco^; and a line locus
) =0
208 DYNAMICS. [CHAP. xv.
for the centre of mass. Eliminating p between this and the second equation,
the result is
V . Vc^A/OV - MJSwjV* V sv >i
or separately,
SA'/V and Swj^Wt
The body has therefore a given moment of inertia (Sw^^a)^ round (Oj, and
a given product of inertia ( - SUco^TJA/) with respect to 'U<D 1 and U A/ ; but
it is otherwise indeterminate.
For two degrees of freedom, the two force equations completely determine
p, and the couple equations give completely ^^ and <o) 2 . There remains
only one unknown constant, the moment of inertia (SVcDjCOg-^Vw^) with
respect to the line perpendicular to coj and co 2 .
The dynamical constants are completely determinate in the case of three
degrees of freedom. Compare generally Art. 122, Ex. 6.]
Ex. 11. Two three-systems of screws can be in one way correlated, so
that each screw of one system, regarded as an impulsive screw, corresponds
to a screw of the other system regarded as an instantaneous screw. (Ball,
Treatise, Art. 318.)
[This has been virtually proved in the last example. We have to show
that if cr^^o) and ju,=< 2 A are two three-systems of screws, it is possible to
design and place a rigid body so that M(cr Ypo>) = A, and </xo = ju, VpA
become identities when o- and //, are replaced in terms of A and <o and when
a one-to-one relation is established between A and o>. Substituting for p,
and o-, we have M($ l -'Vp)u = \ and <o)=(<jk 2 - Vp)A, so that
remembering that <f> is self -conjugate, and this holds for all vectors A. Hence
where Xi an ^ X2 are Hamilton's auxiliary functions for fa and fa. And
because X is perfectly arbitrary, we have (xi + X2 / )/ ) = 2 2i ^ 2i * 8 the s pi n ~
vector of fa fa Thus the vector to the centre of mass is
hence M~ l <$ is expressed in terms of fa and of fa. The two three-systems
are connected by the relation -^(<#>i-2V(x 1 + X2; l 2i) (t)== ^j so tna ^
screw of one system corresponds a definite screw of the other.]
Ex. 12. Screws (/z, A) and (cr, o>) are connected by the relations
where < 15 < 2 , <s and <f> 4 are four given linear vector functions. Find the
conditions that (///, A') should be reciprocal to (o-, o) whenever (//,, A) is
reciprocal to (o-', u>').
[The general relations of this example establish a homography between
screws (/i, A) and (a, a>) ; and when the conditions of mutual reciprocity are
satisfied, the homography is said to be chiastic (Ball).
The conditions are simply
= Sa (<#> 1
where o>, a)', a- and o-' are arbitrary vectors. Putting <o and <o' both zero, it
appears that <j> t must be self -con jugate. In like manner <f> 3 is self -con jugate,
and the condition reduces to Scr^o- <)V)w = So-(< 2 - ^O 03 '? which requires <^> 4
to be the conjugate of < 2 . Thus the general chiastic homography is defined
by relations of the form
A =a faa- 4- <(>&), p = <
where </> x and < 3 are self -conjugate.]
ART. 123.] THEORY OF SCREWS. 2Q9
Ex. 13, The screws of impulsive wrenches applied to a free rigid body
at rest in a given position (or the screws of the reduced wrenches applied to
a constrained body) are in chiastic homography with the screws of the
corresponding instantaneous twist velocities.
[Here A.=J/"(cr-Vp<o), /A=<eD+J/T/D(o---V/3a>) and the conditions are
satisfied. This may be seen still more simply by taking the base-point at the
centre of mass.]
Ex. 14. The united screws of a chiastic homography are co-reciprocaL
[For a united screw ju,=#o-, AtfO), and for a second united screw //=#V,
A' = # V, and hence # S (oV + cr'co) = S (JM>' + or' A) = S (/i'o> + o-A') = 4/S ((/&> -f G-O)'),
so that the screws are reciprocal or else x=x r . In the latter case every screw
of the system (cr-f fcr', <o-Ma>') is easily seen to be a united screw of the
homography. The theory is quite analogous to that of the axes of a self-
conjugate function. The united screws in the general homography are to be
determined by solution of the equations zu^^a- +<$>#>, #cr=< 3 <t)+< 4 <r. On
elimination of o-, we have
4> 3 w = (<#> 4 - x) <}>r\<l>2 - #) >.
Compare Art. 115 (x.), p. 186.]
Ex. 15. There are n real principal screws for every position of a rigid
body having freedom of the nth order, so that the body will begin to move
from rest along one of these screws when a wrench is administered on that
screw.
[For the centre of mass as base-point, if (/*, A) is on a principal screw, we
have ju,=#cr, A=#o> and also /i-/A,=<a> and A-A,=Jfcr. Now if (<r ls o^),
(cr 2 , o> 2 ), etc., are screws of the freedom we deduce from these expressions the
n conditions
#S (o-jco + crcoi) = Sox^o^ + J/Skroj, etc. ;
because the evoked wrench is reciprocal to every screw of the freedom.
Also a>=2tf n G) n and cr=2? n o- 7l , and on substitution for & and a- and on elimi-
nation of tie scalars t, a determinant of the %th order in x is obtained.
Putting x equal to one of the roots of this equation, the scalars t can be
found from TZ- 1 of the conditions.
Just as in the case of self conjugate functions, if a root x is imaginary
? the corresponding principal screw is imaginary
_
and there is a conjugate principal screw (cr'-V-lcr", A/ -"/-I A."). By the
last example these screws are reciprocal, and we find that
must vanish. This cannot be because the energy of a body moving with a
real twist-velocity (cr', a/) or (cr", a>") is essentially positive.]
Ex. 16. A body which is imperfectly free moves under no applied forces.
Find the conditions that the instantaneous screw should be permanent.
[When the instantaneous screw is momentarily stationary it is said to be
permanent (Sir "Robert Ball). For the centre of mass as base-point, the
equations of motion are
where (?? ,) is the evoked wrench. The condition of reciprocity gives
So><a> + J/So-cr=0 ; and for a permanent screw u>=#co, <r=j?cr, and we must
have #=0 because Sox^w+ifcr 2 is essentially negative. By means of the
equations of constraint we can eliminate , and 17, from the conditions
J.Q.
210 DYNAMICS. [CHAP. xv.
Ex. 17. To find the principal and the permanent screws for freedom of
the third order.
[Here <r=fa<i> where fa is a given linear function, and the screws of the
constraint belong to the reciprocal three-system p,, = - fa A,. For a principal
screw
and on elimination of A.,, we see that
so that <o is an axis and x a root of a determinate linear vector function.
For a permanent screw,
and on elimination of , we find
and o> is now an axis of the new linear function < -
In the special case of rotation about a fixed point the principal screws
coincide with the permanent screws.]
CHAPTER XVI.
THE OPEBATOR V.
(i) The Associated Linear Functions.
ART. 124. In Articles 54-57 we investigated some funda-
mental properties of the operator V, and we propose in the
present chapter to supplement and develop the results already
obtained and to illustrate the application of the operator to
physical investigations.* Compare pp. 69-77.
In the first place we shall consider the invariants and the
auxiliary functions for the linear function
(l.)
* Hamilton's writings on the operator V consist, so far as I am aware, and I
have searched through his manuscripts in the library of Trinity College, of a
communication to the Royal Irish Academy (July 20, 1846) which is published
in the Proceedings, Vol. iii., p. 291, and practically reprinted in the Phil. Mag.
of the following year, and of Art. 620 of the Lectures on Quaternions. In the
Lectures he writes : " The bare inspection of these forms may suffice to convince
any person who is acquainted, even slightly (and I do not pretend to be well
acquainted), with the modern researches in analytical physics, respecting
attraction, heat, electricity, magnetism, etc., that the equations of the present
article must yet become (as above hinted) extensively useful in the mathematical
study of nature, when the calculus of quaternions shall come to attract a more
general attention than that which it has hitherto received, and shall be wielded,
as an instrument of research by abler hands than mine." He denoted the
operator by the symbol < . In the Elements the operator occurs in a disguised
form in Art. 418 (v.), V being replaced by -D a where a is the vector operand.
In the first note to Art. 422 of the same volume and in a letter to Dr. Salmon
(Graves J s Life, Vol. iii., p. 194), he announces his intention of concluding the
work with a brief account of a "quaternion transformation of a celebrated equation
in partial differential coefficients, of the first order and second degree, which
occurs in the theory of heat, and in that of the attraction of spheroids." Un-
fortunately the volume was left unfinished at his death.
The applications, predicted by Hamilton, have been made by the able hands
of Tait, as will be seen OD reference to the volumes of his collected Scientific
Papers (Cambridge, 1900), and to the last edition of his Treatise on Quaternions
(Cambridge, 1890). M'Aulay has also made valuable additions to the subject in
his Utility of Quaternions in Physics (Macmillan, 1893), and the note in the
Appendix to the new edition of Hamilton's Elements (Vol. ii., pp. 432-475) may
perhaps be consulted with advantage.
No satisfactory name has been proposed for the operator. The author prefers
to call it Hamilton's delta or more generally delta.
.'-'A fl'/l (
-.ft.vi?
212 THE OPEEATOE V. [CHAP. xvi.
which we noticed in Arts. 112 and 113 in connection with the
theory of heterogeneous strain. See p. 181.
When the points of a field receive a small continuous dis-
placement so that the vector to a point changes from p to p+o-dt
where dt is some small scalar and where a- is a continuous
function of p, the vector p+a to a neighbouring point changes
into p+a+(<r+<j>a)dt The vector line-element a at the ex-
tremity of p accordingly changes into a+<pa.dt. The vector
area-element Va^S becomes V (a + <j>a . d)(/3 + <j>/3 . dt), or, neglecting
the square of dtf, this is (Art. 65 (iv.), p. 91)
* or a
The volume-element Sa/3y changes into
when we neglect the square of dt. If <r denotes the velocity of
the points in the field, varying from point to point, and if dt is
the element of time ; if dp, dv and dv are respectively a vector
line-element, a vector area-element and a volume-element, at the
extremity of the vector p, the rates at which these elements
change are
T> t .dp = <pdp, D{.dv=x /dl/ > D t .d<y:=m"dv; ......... (n.)
and these relations clearly indicate the meanings to be attached
in this case to <p, x and /m "- ^ke scalar m" is called the
divergence and SV<r is the convergence.
Again the small strain at the extremity of p due to the dis-
placement <rdt may be resolved into a pure strain, which converts
a into a+(0+$') a - d, and a rotation represented in magnitude
and direction by edt where e is the spin- vector of </> ; for we have,
if
* = (1 + Ve . d)(a
when we neglect dtf 2 . Hence the spin-vector e represents in
magnitude and direction the angular velocity of the element at
p when <r denotes the velocity of its points in the field.
It remains to exhibit e, m" and ^ in terms of <r. We have for
tny three vectors
V/3y . <f>a+ Vya .
and the first quaternion invariant (Art. 67, Ex. 7, p. 97) is
m"-2e=-V<r, and m"=-SV<r, e=VV<r ....... (ill.)
Further, x = vvv ^ x'a=- v - VaV.o-, ............... (iv.)
AET.J24.] THE ASSOCIATED LINEAR FUNCTIONS. 213
because, for example, x<* = ("-0)a = -SV<r. a+SaV. <r. It is
evident that x an ^ have the same spin-vector. The vector
2e or War has been called by Clerk Maxwell the curl of. the
vector a-.
The function^' and the invariants m' and m are related to
the transformation which converts vectors p into vectors a- where
a- is a given but arbitrary function of p. As in Art. 63, if do-,
div(=Vd(rd'o-), and dv (r (= Sdo-dVdV), are the elements into
which the elements dp, dv and dv at the extremity of p trans-
form, we have dar=<f>dp } d*v=\//di/, dv^mdv. To calculate yfs
in terms of or it is necessary to use temporary marks to associate
the corresponding operator and operand, and we find (p. 90)
WSacr. V'S/3cr' = VVV'Sa<rS/?</.
Now we may also put
r/rVa/3- VV'VSa<r'S/3o-= - VVV'Sa<r'S#r,
so'that on addition,
^Va = J VV V'(SaerS/3o-' - Sacr'S/3cr) = - } V VV'S Vc
or for an arbitrary vector y,
^y=_JVVV / Scr(rV ^V^-tSyW'.Vaxr', ....... (v.)
and in these expressions the accents are to be removed after the
performance of the indicated operations *
Just as in (in.) we find the quaternion invariant of \f/ f
m / -20e= - JVVVToV, .................... (vi.)
remembering that <f>e is the spin-vector of \f/ (Ari 68, p. 98).
Thus m'=-SWV'VoV, ^e=iV.VVVTcrcr / , ........ (VII.)
and this expression for 0e should be verified by operating with
on the value already obtained for e.
It is also a useful exercise to verify that the third invariant is
m=iSVV'V // ScrcrV, ..................... (vin.)
but a more familiar form of this invariant is
dx 'dy 30
i device employed here is quite analogous to a transformation in Aron-
hold's symbolic notation.
214 THE OPERATOR V. [CHAP. xvi.
which is obtained by putting p^ix+jy+Tcz, <r=iu+jv+kw
Ex. 1. Show that in terms of i, j and ,
1 m " =2 3S' m/==
'dv'dw
Ex. 2. In terms of three arbitrary differentials of p and of the corre-
sponding differentials of <r,
. 2dcrSad / pd"p . _2dpSd'crd"cra _Sdcrd'crdV .
* a ~ Sdpd'pd"p > ya ~ Sdpd'pd"p ' m ~~ Sdpd'pd"p '
2dcr.Yd'pd"p , ,
= -
(a) If d^ss^^or, write down the corresponding functions for <j> a and find
the relations between them and those for d>.
Ex. 3. Prove that
'du 'bu_ "bu_
"bx 'by ^bz
*bw *bw *bw
*bx 'by 'bz
*bx 'bx 'bx
*bu 'dv 'bw
*by 'by
'bz_ 'bz_ 'bz
026 c/y ow
= 1.
Ex. 4. If <r, a vector function of p, satisfies a scalar equation /(<r)=0 for
all values of /), the third invariant m of the function < vanishes ; and con-
versely if m vanishes <r satisfies an identical relation.
[If do- is the differential of o- corresponding to an arbitrary differential
of />, we have d/((r)=0 or (say) Sjudo-=0. Hence the three differentials
of <r corresponding to three arbitrary differentials of p are coplanar and
Sdo-d / <rdV=0. Conversely, if m is identically zero, three differentials of a-
corresponding to three arbitrary differentials of p are linearly connected, or
Zdo--h*d / o-+rdV=0, suppose. Hence <r can receive only two independent
variations, or a relation of the forrn/(o-)=0 must be satisfied by cr.]
Ex. 5. If <r satisfies two scalar relations / 1 (o-)=0 and / 2 (<r)=0, the
function ^r must vanish, and conversely.
Ex. 6: -If /(<r)t=0, and if we write d/<r=S/Adcr, we shall have <>=0.
Ex. 7. If cr is a function of p and if d<r=<f>dp, prove by comparing the
operators d= -Sd/>V= - SdcrVo-, that
VH/V*
where 't^o- operates on a function of cr in the same manner as V operates on
a function of p. (Taitfs Quaternions, Art. 480.)
Ex. 8. If <d/> is the differential of a vector function of p,
where a is an arbitrary constant vector ; and if it is possible to find a scalar
multiplier to render <dp the differential of a vector function,
[Note tliat c/> f a= - VSacr if <dp=dcr.]
ART. 125.] lOTEGBATION THEOREMS. 21$
Ex. 9. If GI and <7 2 are the principal curvatures of a surface u= const.,
show that ^ + Ct _ _ svu y^ c ^ = _ jsVWVUVwVV.
[See my note, Elements, Vol. ii., p. 251. If TJ and r 2 are tangents to the
two lines of curvature,
and (Ex. 4), since TUV^=: 1, the third invariant of the function - SdpV . TJVw
is zero, and (7 1} <7 2 and zero are therefore its latent roots.]
(ii) Integration Theorems.
ART. 125. It has been shown in Arts. 55 and 56 that the
form in which the operator V naturally presents itself leads to
the two results (pp. 72 and 73).
;. V).gr; ............. (i.)
the first integral being taken over a small closed surface of
which dj/ is an element of outwardly directed area while dv is
the included volume ; and the second integral being taken along
a small plane closed curve of directed area dv, where rotation
round dy in the direction of the circuiting is positive. In both
relations q is a quaternion function of the variable vector p.
In order to extend these results to integration over finite
regions, we shall first suppose that the quaternion q satisfies
certain conditions : (A) that it is free from discontinuity, (B)
that it is single-valued, (c) that it does not become infinite at
any point of the region. Further we suppose (D) that the region
included in the surface over which we propose to integrate is
simply connected, so that any closed circuit drawn in that region
can be made evanescent by continuous variation without cutting
through the surface.
On these suppositions, we divide the region within a closed
surface into infinitesimal parallelepipeds, and we apply the
theorem of Art. 55 to each. Adding together the integrals
Jdi/ . q over the faces of these parallelepipeds, the sum obtained is
equal to the stun of the corresponding elements Vq . dv ; but over
an interface corresponding to two parallelepipeds the directed
elements are opposite, so that if one parallelepiped contributes
an element dv . q, tlie other contributes an equal and opposite
element dv . q ; consequently the sum of the integrals Idy . q is
the integral over the bounding surface. Moreover the sum of
the elements Vq . dv is the integral I Vq . dv throughout the
volume, and we have Idi/. g= iVg. dv, ............................ (ii.)
216 THE OPERATOR V. [CHAP. XVL
where the first integral is taken over the surface and the second
throughout -the volume.
Tinder the same conditions we can fill up any continuous
closed curve by a net-work of parallelograms described on any
surface terminated by the curve, and if these are all circuited in
the same direction the elements contributed by the common sides
cancel, and fdp.ff=fv(di/. V). S> ..................... (m.)
_J --- uL~~- - -
where dp is a directed element of the curve and dv a directed
element of the surface. Hence it follows because (in.) has a
value independent of any particular surface through the curve
that over any closed surface
.......................... (iv.)
(A) Suppose a surface to exist over which q is discontinuous, and imagine
the region of the volume integral to be divided into two regions by the
surface of discontinuity. Applying (IT.) to each of these regions and adding,
we find
fc-22) ........................... (v.)
an element of the surface of discontinuity furnishing the parts
dv^.ft and di/ 21 . 2 , or d^^-^).
(B) If q is not single- valued, it is not hard to see when infinite values of
Vq are excluded from the region that, assuming any one of its values for q
at any point of the region, the value of q at every other point of the region
is determinate. In fact starting from a point p with a given value of q we
can return to p with a different value only if we thread some circuit along
which q is indeterminate ; and if q is indeterminate anywhere within the
region, its corresponding deriveds must be infinite, which is contrary to
supposition. When a curve locus of indeterminate values of q exists in the
region, we may enclose it in a tube and so isolate it from the region. The
region thus becomes multiply-connected (D).
^ (c) If q becomes infinite at any point, we exclude that point by a small
sphere concentric with it and we take account of the surface integral over
the sphere, the vectors representing the elements of directed area being
drawn outwards from the region, that is, towards the centre of the sphere,
and the radius of the sphere being ultimately reduced to zero.
Taking the origin at the point, the element of directed area over the
surface of the sphere is di/= - Up . r*. dli if r is the radius and dO an element
of solid angle. Then for the sphere
Jav.g^-JaO.U/).^ ............................. (vi.)
If over the surface of the sphere
q=qo+r~ l . qi+r~*. 2 + *- 3 . 3 +etc., ................... (vn.)
the surface integral vanishes unless q 2 exists, and it generally becomes infinite
or indeterminate if q 3l etc., exist. Of paramount importance is the case in
^hich q contains the term VTp- 1 . <?= - Up . Tp~ 2 . e. In this case if no higher
negative power of r occurs, the integral becomes
..... Jav.gr* -Jaa.e-4ir, ........................... (vm.)
ART. 125.] SINGTJLAKITIES IN INTEGRATION. 217
and we must replace (n.) by
^jVg^. d^jdv . g - 4?re, (ix.)
the origin being excluded from the volume integral.
In general when q%, etc., are zero, by a well-known theorem in spherical
harmonics (Art. 127) we need only consider the terms in q% which are linear
in Up and which we may take to be SaUp+<Up. Writing TJp^li+mj+nk
where I, m and n are the direction cosines of U/>, and remembering that
JdG . Z 2 = ^TT, JdQ . Im 0, etc.,
we have
where m" is the first invariant and where e is the spin vector of <j>. Accordingly
we must in this case replace (n.) by
where the integral on the right is taken over the boundary and where the
remaining terms are contributed by the surface of the evanescent sphere.
(D) If the region is multiply-connected we render it simply connected by
drawing diaphragms* when we fall back on case (A) if q happens to be
many-valued. A diaphragm corresponds to a surface of discontinuity, and
q 1 q% in (v.) becomes np where p is the cyclic increment of q and where n is
an integer.
Considering now the similar cases of exception for the circuit integral, we
shall suppose
(A') that a surface of discontinuity cuts the given circuit in two points
A and B. Let the surface containing the mesh- work be drawn through an
arbitrary curve ACB on the surface of discontinuity. On adding the results
of integration for the two circuits consisting of the part on one side of the
surface of discontinuity and the curve ACB, and of the part on the other side
of the surface and the curve BCA, we have exactly as in (v.)
It follows from (TV.) that we get exactly the same result had any other
curve ADB been taken on the surface of discontinuity.
(B') If q is not single-valued over the continuous net, its value is definite
if a definite value is chosen at some one point of the net, or else q is inde-
terminate at some point of the net. Such a point may be surrounded by a
small closed curve joined by a barrier to the circuit. The barrier must be
treated as a line of discontinuity and the value of the integral round the
closed curve must be taken account of.
(c') When q becomes infinite at a point on the surface of the mesh- work,
let the point be surrounded by a small circle of radius r. Then the relation
becomes, when we exclude the point from the surface integral,
JYdvV. q=> Jdp . q JrdU/o . (qQ+qtf~ l +q# r *+*te.\ (xm.)
the second line integral bekig taken round the circle.t This integral
vanishes unless there are negative powers of r. The part depending on q l is
JdU/> . q l = jdUyo . (SaTJ/> -t- <TJ/>)
*The interior of a hollow curtain ring becomes simply connected when a
diaphragm is drawn across one normal section.
t The two line integrals are taken in the same sense of rotation round the axis
of the small circle. If we choose the minus sign, may be j>laced on the right of
the sign of integration, and then we shall have the surface integral equal to the
sum of two line integrals taken in opposite directions.
218 THE OPEKATOR V. [CHAP. xvi.
suppose where < is a linear vector function, the terms not linear in Up
leading to a vanishing integral round the circle. Putting Up = i cos w +j sinw
where i and / are in the plane of the small circle, the integral easily reduces
to ir(jSai-iSaj+j<l>i-i<l>j\ and to fl-(Va&-x'+2S&) where x' and have
the same signification as in the chapter on linear vector functions.
Ex. 1. If /(V) is any linear function of the operator V with constant
coefficients,
and J?./(dv)=j
[No step in the proof of the simpler case need be modified. In the
second set of relations the operator is placed in front of the operand. See
Art. 57, Ex. 11, and M'Aulay's Utility of Quaternions in Physics?]
Ex. 2. In general if /(a) is a linear function of an arbitrary vector a
while the variable vector p is involved in the constitution of the function,
show that
where /(V) means that V operates in situ on the variable vector p as involved
in the structure of the function.
Ex.
where no infinites occur.
[See Tait's Quaternions, Art. 504. Here the line integral is JVdpVTp" 1 ,
which transforms into
3. Prove that J ^= - JSdvV. VT/r 1 ,
JV.YdvV.VTp~ 1 or J dvV^p-^J Sdj/V . VTp- 1 .]
Ex. 4. Prove that
J 2dt>=! J p . Vq. dv-
[This is an example of an extensive class of transformations depending on
the invariantal properties of V. Transforming the surface integral, we
have Jpdv=Jp(V)2dv, where V operates both on p and on q. But
pV = Vp = - 3. See Art. 132, p 235.]
(iii) Inverse Operations.
ART. 126. We shall now establish general solutions for the
equations
Vp = q, and V 2 r = g, ........................... (j.)
where q is a given quaternion function of p ; or we shall assign
definite interpretations to the functions
p = V~ l q and r = V" 2 g ........................ (n.)
for all points of an arbitrarily selected region within which
infinities do not occur.
ART. 126.] INVERSE OPERATIONS. 219
We shall first prove the transformation*
I Vu . Vp . dv = Id]/ . u . Vp I Vu Vp . dv
== JVu . dv . p- jV^-Tp-^V . p d^-47rp (m.)
in the case in which p does not become infinite within the region,
while u tends to the value Tp" 1 at the origin which we suppose
to be taken within the field of integration, and where 4xp in the
third member is 4x times the value of p at the origin. The
suffixes are intended to indicate that the affected symbols are
free from the operation of V.
Surrounding the origin by a small sphere and supposing (V)
to operate in situ on u and on p we have
J Vu . Vp . dv = | (V)u . Vp . dv \ Vu Q Vp . dv
= Idi/ . i& . Vp I
. dv
for the region between the small sphere and the boundary, the
surface integral over the sphere vanishing by the last Article
(compare (vn.)). But these integrals may be extended through-
out the entire region, for we shall show that the integrals taken
through the volume of the small sphere tend to zero when the
radius is indefinitely diminished. Within the sphere we may
take u-Tp" 1 and dv = Tp 2 . dQ . dTp,
so that (Vt&.Vp.dv=-jUp.Vp.d].dTp
which vanishes in the limit. A fortiori the integral
Vu Vp . dv = Tp - 1 . V*p . dv
\
for the small sphere vanishes. Thus the first part of (in.) is
proved.
Again for the field exclusive of the sphere
I Vu . Vp . dv =s V-w, . (V)p . dv \VuV . p dv
= I Vu . dv .p4nrp I VuV . p dv
by (vm.) of the last Article because for the surface of the sphere
. dy ,p= + Tp- 2 . Up . Up . Tp 2 . d2 . p= -
*It is manifest from the proof of these relations that they are valid when
neither p nor u become infinite in the field of integration provided we omit the
term in Tp' 1 and the term 47rp.
THE OPEEATOE V. [CHAP. xvi.
Also it is easy to see that VTp^V^V^p' 1 (or more generally
that V . Sq .V = V*Sq), and this vanishes for all points of the
region outside the small sphere. And because u Tp~ l does not
become infinite at the centre of the sphere, we have
| Vu . Vp . dv= I Vu . dv . p 4mp I V(u Tp" x )V . p . dv,
where the volume integrations are extended throughout the
whole of the original region. Thus (in.) is completely established.
In particular when u is a scalar we may replace (in.) by
I VuVp . dv Idy . i& . Vp \uV' 2 p . dv
= (Vu . dv.p- jVv-Tp- 1 ) .pdv-fap. ...(m.)'
Changing the origin or replacing p by p 7 p in (in.) 7 , and
supposing p 7 to be the current vector in the integrations, we
obtain for the particular case in which i& = T(p p 7 )" 1 the
important identities,
V 7 yW
the second
being deduced from (ill.)' by replacing Vw by
- 1 or by its equal -VT(p 7 -p)- 1 and taking V out-
side the sign of integration.
If then V = , we have
q'dv'
and in this relation p' is any function which over the boundary
satisfies Vp=q.
In like manner
where r is any function which over the boundary satisfies
V*r==j. It may be observed that in these results there is a
certain analogy to the solutions of the linear function equations
of Art. 65, p. 92.
. 126.] INVERSE OPERATIONS. 221
If we operate on (vi.) by V and' put p=Vr we find on com-
parison with the second form of (v.) that
V . V- 2 = V- 1 ^ ...(vii.)
because the last integral of (vi.) vanishes under the operation
of V (or of V under the sign of integration operating on
VT(yo' yo)" 1 ) provided /> does not terminate on the boundary.
Ex. 1. Find the potential which produces a given distribution of force
in a given field.
If is the force and P the potential, we have to determine a scalar
function P from the equation = - VP. ' By (v.) this function is
SV'g.d*' f Sdy? /^SdvV-.T(p- P r 1 1
'rJ 4ir J
Ex. 2. A quaternion p which satisfies the equation V 2 p=0 throughout a
given region is expressible as a surface integral over the boundary ; and a
quaternion^ which satisfies Vj0=0 throughout the region is of the form
dv'.p'
4,rT(p-p'y
Ex. 3. A scalar satisfying the equation VP=0 is constant. A vector
satisfying V<r=0 is expressible in the form aVP where P is a scalar
function satisfying V 2 P=0.
Ex. 4. Construct quaternion functions of p, homogeneous and of the
first and second orders, which shall vanish under the operation of V.
[For the quadratic function assume j0=Sp$ p+2a n Sp< n p where w==l,
2 or 3. We have Vp= 2< p - 22< n pa n , and if BVp is identically zero the
condition 2<^> n a n =0 must be satisfied. In order that VVp may vanish, we
must have <oP SV^oan^ +2< n Vpa n since ^> i s self -con jugate. Again,
because V 2 jp=0, the first invariants of the functions < must vanish. But in
general w' / Vpa=Vp<a+V<pa + <Vpa, and in the present case
Hence by the former condition -2V< n p<Xn is a self -conjugate function
provided only that 2< n an=0, and that the first invariants are zero. Thus
where 7^=0, 2^(^=0, vanishes under the operation of V.]
Ex. 5. Determine the extent of the arbitrariness in the dissection of a
quaternion -into the parts V~ 1 SVg r and V" 1 Wq on the supposition that
v~*8Vq is a yector.
[The most general expressions for the parts are V^SVy+cr .and
--o-, where <r is a vector satisfying Vcr=0. See Ex. 2.]
T.TT. 6. Divide a vector cr .into two parts o^ and <r s so that SVari=0 5
W(T 2 ==0.
[Here cr 2 =V' 1 SVcr and cr 1 =V"" 1 VVo'. We may calculate one of these,
say <r 2 by the. general formula, and^the. other is 07- cr^]
222 THE OPEEATOE V. [CHAP. xvi.
Ex. 7. The general solution of the equation
may be written in the form
[The equation may be transformed into (m+^)VSVo-+7&VVVo-=, and
by the last example, VSV<r=(*+*)~ 1 V- 1 SV > VVVcr=7r 1 V- 1 VV The
solution given above of the equation of equilibrium of an elastic solid may
be expressed more simply in the form cr='
Ex. 8, If V 2 jp=0 at all points within a closed surface, and if V^r^O at
all external points ; if p,=p over the surface and if p t tends to zero at infinity,
r
P ~ J
[Integrating throughout external space we find if V^>,=0, see note p. 219,
when p terminates at an internal point so that T(/>' p)" 1 does not become
infinite. The surface integrals are to be taken over the closed surface and
over an indefinitely large surface, but it easily appears that the latter part
of the integer vanishes since p 4 vanishes at infinity. Putting V z p=Q in (iv.),
remembering that p,=p over the closed surface, and subtracting, we have
the required result.]
Ex. 9. If f n p is a homogeneous function of p of order n satisfying
V 2 / n /)=0, show that
when T/D<a, the integration being extended over the sphere whose centre is
the origin and whose radius is a.
[The function corresponding to the p t of the last example is
(See Art. 57, Ex. 12.) Here V(p'-p;)=(2n+l)a- l Up' .f n p' over the
sphere and di/=UpTdv'.]
(iv) Spherical Harmonics.
ART. 127. If / n (V) is any rational and integral function of V,
homogeneous and of order n, the function f n V .Tp~ l is a solid
harmonic of order (TZ+!), for it is a homogeneous function of p
which vanishes under the operation of V 2 , the scalar operator V 2
being commutative in order of operation with / n V. Further
Tp^./nV.Tp- 1 is a solid harmonic of order n. (Art. 57,
Ex. 12, p. 76.)
Because we may suppose f n V to be expanded in the form
/nV^SaS^VSaaV..^^ ..................... (l.)
it follows from Art. 54, Ex. 2, p. 70, that
./ n V.T P ^=(-)-. 1.3.5 ..... (2^- l)(f np - V./*-2/>), (n.)
ART. ir.] SPHEEICAL HAHMONICS. 223
where f n - z p is a determinate function of p, homogeneous and of
order 73, 2. Hence we may expand any homogeneous function
of p of positive order n in a series of solid harmonics, of orders %
7i 2, 7& 4, etc.,
where / n _2/5> /n-4p, etc., are functions defined by equations such
as (IL),
Any integral of the form P = \p&v.T(p (0)- 1 in which co is
the current vector and in which p is independent of p may be
expressed in the form
P-/V.T/.-I,
provided Tp is not less than the greatest of the tensors Tew.
For (Art. 59 (XL), p. 79),
and we may speak of P as the potential at p due to a distribution
of density p -although it is not necessary to suppose that p is a
scalar.
If Q = \q<iv'. T(fc>' p)" 1 is the potential of a second distribution
of density q, the mutual potential is
If the second distribution lies outside a sphere of radius a
having its centre at the origin and including the first distribution,
we have by (v.),
'
provided we reduce the temporary vector p to zero after the
performance of the operations indicated, and the suffix serves
to remind us of this reduction.
If Q=g n (p) is a solid harmonic of positive order n, and if we
suppose the corresponding distribution to be a surface distribu-
tion on the sphere, we may replace
gdt/ by (4T)- 1 . (2^+ 1) . a- 1 . ft (0 . Td/,
224 THE OPERATOB V. [CHAP. xvi.
or by (4T)" 1 .(2
utilizing Ex. 9, Art. 126, and dropping the accents as being no
longer necessary. In this case (VIL) becomes
co).dn ....... (vm.)
In this expression it is only necessary to take account of terms
of order n in /( V), for g n (p) vanishes under the operation of
terms of higher order, and the results of operation of terms of
lower order vanish when p is reduced to zero.
If P is a solid harmonic of order n 1, the form of the
function /V is given by (ill.), and
and accordingly (vm.) becomes
while if the order of the harmonic P is (m+1) where m is not
equal to TI ? we have
(XL)
Again if
P = T(p-a)-* = P v .Tp-i=IXa n Tp-*- l A n (Up), ...... (XII.)
find on substitution in (vm.),
...... (xm.)
because /( - V) . g n (p) = <T SaV . g n (p)=g n (p + a).
Hence we can expand any function <7(U/>) in a series of
spherical harmonics, the harmonic of order n being
.dQ ............ (XIV.)
Ex. 1. A scalar solid harmonic of order -(+!) may be expressed in the
form SajV . SctjjV ..... Sc^V . T/r 1 ,
where c^, a 2 , ... a^ are real vectors.
[Consider the edges common to the cones F n p=Q, /> 2 =0. These group into
conjugate pairs P+*/^lB' and /3-J^I/3 f , and each conjugate pair lies in
a real plane Sap=0 where a=Y/3/J'. Having determined the vectors
... a n we have a relation of the form
AET. 128.] SPHERICAL HAEMONICS. 225
where t is a scalar and where F n -%p is a homogeneous function of p of order
n 2. If F n V is the generating operator (see (ix.)) of the harmonic we have,
on putting V for p in the above relation,
F n V . Tp-W . SctiVSagV ... SanV . Tp- 1 because V^T/tr^O,
and the scalar t can be found by comparing a coefficient.]
Ex. 2. If q is a quaternion associated with each element of mass of
a body,
where r is the vector from a point in the body to the element d?rt, where q Q
is the value of q at the origin of vectors T, and where Vv operates on/(V) as
if it were a function of a vector V.
(a) The first terms of the f unction /(V) are
where M is the mass of the body, T O the vector to the centre of the mass, <
the inertia function for the origin of vectors T and J, -5, C the principal
moments of inertia for the same point.
[We have
= J(l - SrV +|SrV 2 -etc.)dm . ? ;
and because SrV 2 =T 2 V 2 +YrV 2 , JVrWrdm=4>V
and Jr 2 dm= -(4+5+0),
the expansion is justified. Again the differential of /a corresponding to da is
d/a= - SdaVa ./a= - JSdcmT STa dm.]
Ex. 3. A heavy body is placed in a field in which the gravitational
potential is P. The potential energy of the body (F), the resultant force
and the resultant couple (X. and p) acting on the body and referred to its
centre of mass, are
. VP, /*=V<1>W . P.
(v) Various expressions for V.
ART. 128. We shall now examine in greater detail than in
Art. 57 the various analytical expressions for the operator V
and for V 2 .
In terms of three arbitrary differentials we may write
V=Ad+X'd / +X / 'd // , ........................... (i.)
where (Art. 54 (VL), p. 70)
Vd'pd'p ,_ Vd'pdp Vdpd'p ( .
"" A ~~Sdpd'pdy A " Sdpd'pd'p ^ ;
The operator V 2 is now
V 2 = 2X 2 d 2 +2(X / X // d'd // +X // X / d // d / )+2VX . d, ......... (m.)
and in the third sum V operates on the vectors X alone and not
on the operand of V 2 .
J.Q. P
226 THE OPERATOR V. [CHAP. xvi.
Kemembering that V 2 is a scalar operator, this breaks up into
the two parts
; ......... (iv.)
(v.)
It is only when the differentials are independent that the
order in which the differentiations are performed is indifferent.
and it is only in this case that we can generally suppress the
terms involving d'd" d"d' and similar expressions in (v.).
When independent differentials are employed, we use the
expression (Art. 57. (in.), p. 74),
or as it may be briefly written
where the vectors v l9 i/ 2 and j/ 3 satisfy the relations
Sl/ i/>i + 1 = ete - Sj/g/og = 0, S^ 3 p 2 = 0, etc. ; ...... (vm.)
or again we may put
as we see by comparing the results of operation of the forms
(vn.) and (ix.) on u, v and w. Thus
i^srVw, j/ 2 =Vv, j/ 3 =V^ ..................... (x.)
and VVi/^0, VVj/ 2 =:0, VVj/ 3 = ................... (xi.)
The vectors v 1? v 2 and j/ 3 are the normals at the extremity of />
to the three surfaces u = const, v= const, and w = const, which
pass through that point.
The appropriate expressions for V 2 are now
.; ........... (xn.)
or V^2(V^. + 2SV,V W . + EV^.A. ...(xm.)
Again introducing the operand q for the sake of greater
clearness, we may write
Vg=-
because the terms which involve the second deriveds of /o, such as.
ART. las.] VAKIOUS EXPEESSIONS FOE V. 227
^PwPs ?+Vp 3 yo 12 . q, cancel in pairs. Operating with this form
of v on (vi.), we have
where the second sum contains six terms, and to this the sign
S may be prefixed. Or in terms of the vectors v it easily appears
that this reduces to
Whenever the surfaces, u = const., v = const, and w = const., are
equipotential surfaces with the corresponding potentials, u, v
and w, the operator V 2 is a homogeneous quadratic in the
o o *~\
differentiating symbols , , . This property follows
directly from (xin.). The converse is also true.
When the surfaces are mutually rectangular, the operator V 2
is independent of the products of differentiating symbols. In
this case we find from (xv.) the most convenient expression for
V 2 to be i
T"TO * ^T> '
f7TTT---^V- *:,) (XVIL)
Pi
Ex. 1. Determine expressions for V 2 where
(1) p = u {(i cos w +j sin w) sin v + k cos v } ;
(2) p = u (i cos v +j sin v) + few ;
(3) p= Av /{(<^H-w)(^>-i-'z;)(<^+w)} . e, as in Art. 84.
Ex. 2. If a scalar function P of a scalar function w of p can be found to
satisfy V 2 P=0, show that
(Vw) 2 . ^ 2 + V 2 w . ;~ =0 and VVwV .
[Sea (xin.) for the first condition. The second expresses that V 2 w . (Vw)~ 2
is a function of u.~\
Ex. 3. Given that a family of surfaces u const, is an equipotential
system, show that the potential corresponding to u is
[See the last example.]
Ex. 4. A family of concentric, similar and coaxial quadrics compose an
equipotential system. Show that the sum of the reciprocals of the squares
of their principal axes is zero, or else the quadrics are spheres. Determine
also the corresponding potentials.
[Here M=S/><p, Vu<j>p t V%=m". The condition of Ex. 2 becomes
V^/
228 THE OPEBATOB, V. [CHAP. xvi.
Ex. 5. Find the condition that the family of surfaces /(p, w)=0 should
form an equipotential system, and determine the potential when the
condition is satisfied.
[Imagine u to be expressed as a function of p by solution of the equation
/(p, u)=0. On this understanding we may treat /(p, %)=0 as an identity
and equate to zero the results of operating on it by V and V 2 . We find
where V operates on / as if / were a function of p alone, and where
consequently V and 4- are commutative in order of operation on f.
Utilizing the results of Ex. 2 to eliminate V 2 ^ and eliminating Vu we find
3 , aP 3 . fdf 1 \ V 2 /
The condition to be satisfied is that the right-hand member a function of
p and u should reduce to a function of u alone by aid of the equation
f(p 9 #)=(). If P(p y u) reduces to a function of u alone by aid of the equation
/(/>, *)=0, ^ must have VF+Vu . |^ || Vw || V/ or simply
Thus the condition required is
Ex. 6. Show that the family of confocals S/o(<^+w) -1 p+l=:0 is an
equipotential system, and determine the potential.
[Here we have V/= -S^+tt)- 1 ^ and | -(^+)-y= -i(V/) 2 ;
also v s /= -
These give
_, p
Ex. 7. The condition that the family of surfaces /(p, w)=0 should
compose a system of characteristic surfaces in an optical medium of constant
[Hamilton's characteristic functions satisfy the relation TVQ=n, where n
is the index of refraction of the medium. If the family of surfaces satisfies
the condition we must have Q a function of w, so that VQ=Q'Vu= -*Sy v ~ 1 V/,
where the accents denote differentiation with respect to u. Hence when n
is constant, TV/./" 1 must reduce to a function of u y or YV/V(TV/./- 1 )=0.]
(vi) Kinematics of a Avformoble system.
iffere
(l.)
ART. 129. If q is any function of p and t t its total differential
may be written in the form
ART. 129.] KINEMATICS OF CONTINUOUS MEDIUM. 229
and in particular when we replace dp by a-dt we shall write
When cr denotes a velocity, D t q is the rate of change of the
quantity q regarded as associated with the moving point. On
the other hand q is the rate in change of q at a fixed point, and
SdpV . q is the change in the value q from the extremity of p
to that of p+ dp at a given instant.
If dp, dv and dv are elements of directed line, directed area
and volume respectively, at the extremity of p in a medium
moving with the velocity cr, we have by Art. 124 (n.), p. 212,
(ni.)
where* (Art. 124 (i.) and (in.))
r^oO
because for example we have St?D i di/ = STy'di/ = S;(C7dj/.
In terms of the spin- vector e = JVVcr, the divergence
772," = SVcr and the self -conjugate part <j> of $ we may also
write
or explicitly in terms of cr we have
2 = BT - V VV<TGT - crSVtf, g = - VS<757 - V<rVV57 (VI.)
To prove these results observe that
=57 So-(V) . r+Ssr(V) . Q c
and that
where (V) operates i?i situ both on cr and or and where CT O and cr
are free from the operation of V.
In addition we may write
(D t + m")q = q So-(V) . q (vii.)
because this expression is 5 S<r V . g S VV . g .
We may connect this with previous results by observing that
is a consequence of (iv.) where &> is any vector function of p and .
Also SV^=(D < +m' / )SV CT (ix.)
* See H. A. Lorentz, MncyUopadie der math. Wiss., V^ P* 75.
230 THE OPERATOR V. [CHAP. xvi.
We may also observe that if w = VVtrr, we have by (vi.)
since t and p are independent, so that the order of operation by
V and of partial differentiation with respect to t is indifferent.
Hence VV = o), if co = VVcr ....................... (x.)
From these relations we derive various forms for equations of
continuity ; and the voluminal, the areal and the linear equations
of continuity are respectively
(XL)
The first asserts that qdv does not change for the element of
volume; the second requires Ssrdv to remain constant for all
vector areas dv, and Strrdp remains unchanged if g = 0.
Instead of supposing the quantities q, or and cr to be functions
of p and t, we may take them to be functions of t, u, v and w
where u, v and w are three parameters which individualize the
moving point.
This is Lagrange's method, and Euler's method is that in
which everything is expressed in terms of p and t. The total
differential of q we shall now write in the form,
and following the moving point we have
since u, v and w remain unchanged. In particular
=5e
The vectors a and g- now become
*-f -^IS = 1~ VS 1K (xiv.)
as appears on reference to (iv.). The appropriate form for V in
these relations is that given in Art. 128 (vi.) or (xiv.). The
element of volume is now Sfrp^du&v&w, and the voluminal
equation of continuity is simply (compare (in.))
qSpiP2Ps= const (xv.)
f Ex. 1. If c is the density of a continuous distribution of matter moving
with the velocity cr, Euler's equation of continuity is
c = S V (ccr) or D t c = cS Vcr ;
ART. 130.] EQUATIONS OF CONTINUITY. 231
and Lagrange's equation is
- cS/> 1 /o 2 /) 3 = 0= const.
(a) Hence I), log c = - ^ log Sptftfs = S V 2| = S Vcr.
Ex. 2. Show that
cr=<r-<rSVcr, ^=cr- Vo- 2 -Vo-VV<r=I),cr- JV.cr 2 .
Ex. 3. Show that 5p . p = _ y |i
o r ot
Ex. 4. In general
Vgtrr'4-VG7g'=#'', V^'+VGr#'=w'W'+ig'' where o>"=V5rcr'.
[These relations follow most easily from (iv.).]
ART. 130. The integral
^-[Scrdp (i.)
taken from one point to another along a curve depends generally
on the nature of the curve; but if VVs7 = 0, so that sr = VP, the
value of the integral is simply the difference of the values of P
at the extremities of the curve. This integral may be called the
flow of the vector or along the curve.
The time rate of change of F as the curve moves with the
medium with velocity a- is
D^= -JSgdp, (n.)
and if this integral is independent of the nature of the curve,
= VQ, 6-V<rVVcr=V(S<rar+Q), Dcr=V(S<rcr +Q) (in.)
are different forms of the condition to be satisfied, Q being a
scalar function of p and t Other forms of the condition are
VVg = 0, VVBj-VW(rWBT=0, VVD^=VV / VS(7CT / ; (iv.)
or again (Art. 129 (x.))
= 0, where w = Wcr (v.)
As regards the third of (iv.), note that VV 2 Scrcr = 0.
In general we have (Art. 129 (vi.))
DtF= - [S(d- VcrWBy)dp-[S(rar], (vi.)
and
DF= - fs6d/>d~ fsWcrdi;-[S(rST]dt where dv^Vcrdpdt (vn.)
and where [Scror] denotes the difference of the values of SO-ET at
the extremities of the curve. The expression for T>F shows the
meaning of the various terms, dy being an element of the area
swept out by an element of the curve in the time dt
232 THE OPEKATOE V. [CHAP. xvi.
In the case of a closed curve, the circulation of the vector or
in the curve and its rate of change are expressed by
gdp; (VIIL)
or when or does not become infinite at any point of a surface
drawn over the circuit, we may transform the circulation into a
surface integral so that
(ix.)
= - fSwdi/
The circulation is therefore the flux of the vector ft>( =
through the circuit, and the rate of change of the circulation is
the flux of the derived vectors ( = Wg) or the circulation of g.
For any small plane circuit, the circulation S VVzrydv is the
projection of War on the normal to the circuit into the area of
the circuit. Thus WET determines the aspect of the unit circuit
in which the circulation is a maximum, and it likewise gives the
magnitude of the circulation TVVsr in that principal circuit.
In like manner o> determines the aspect of the circuit in which
the rate of change of circulation is a maximum as well as the
value of that maximum.
The vector DjVVtj determines the rate of change of the
circulation from one principal circuit to another following the
motion of the medium. A principal circuit does not generally
remain a principal circuit. We note that by (iv.) and by
Art. 129 (iv.)
<b = WDtfr - Y V'VScro/ - D V Vcr - V W<rWV ; ...... (x.)
and in general we have
(DtV-VDO.^V'Ser'V.j, ................. (XL)
because D,V . q = V q - S<rV . Vq, VD# = Vq - ( V) S<r V . q.
If a tubular surface, drawn through any circuit, is composed
of curves satisfying the differential equation
0; ........................... (xii.)
or, what is equivalent, if
SVsydv = ........................... (xni.)
over the tubular surface, the circulation in any evanescible*
circuit traced on this surface is zero. In particular if ABC
and A'B'C' are two circuits embracing the tube, the circuit
ABCAA'B'C'A'A is evanescible and also the circuit AA'A. From
this it follows that the circulation in ABCA is equal to that in
A'B'C'A', being opposite to that in A'C'B'A', Hence the circulation
* An evanescible circuit may be reduced to zero by continuous variation.
ART. isi.] CIRCULATION AND FLUX. 233.
is the same in all circuits drawn on the tube so as to embrace it
once.
The flux of the vector TS through a given surface bounded by
a given curve is r
........................ (xiv.)
and the condition that this should depend only on the bounding
curve is that the divergence of sy should vanish, or
SVcr = 0, ........................... (xv.)
as we see by transforming the integral over a closed surface into
a volume integral.
The rate of change of the flux is
(xvi.)
and the condition that this rate of change should depend only on
the bounding curve is
SV = or SVsr-S(V)cr. SVsT = 0, or (D,+m")SVsr = 0. (XVIL)
In any case in which SVzrr = 0, if a tube is constructed of the
lines Vd / osr = through a circuit, the fluxes across all sections of
the tube are the same, and the value of the flux is the strength
of the tube. For a small tube we have, if Tdv is the area of a
cross section and if dn is the strength,
Tdi/Ts7 = d<n,, where SVw = ............. (xvm.)
Ex. 1. If VvDacr=0, the circulation of the vectors cr in any circuit
moving with the medium remains unchanged.
[See (in.) and (iv.). We have D,cr= V
Ex. 2. Show that in Lagrange's method
ART. 131. In Art. 126 we showed that any vector to can be
expressed in the form (see (iv.), p. 220)
ar = Vp, (SV^ = 0), ........................... (i.)
where p is a certain quaternion. We shall examine how this
quaternion is related to the flow and the flux of the vector ET.
In terms of p,
(n.)
because S. VSp.dp = dSp. Hence for a closed circuit,
circulation depends merely on Vp. If the circulation" in evlry
circuit vanishes, the quaternion p reduces to a scalar, as we
already observed. The circulatio
circulation in general is expressible
234 THE OPERATOK V. [CHAP. xvi.
We have also
.(VVp-VS(rVV^~(7V 2 Vp)dp + [D,Sp]; ..... (iv.)
and & and or are
g = Vjp-VSerVj)-V(rV*V2?, # = Vp-VVVcrVp-o-V^. (V.)
The flux is
(y=-fSBrd^=-fsdi/V.Sp-fsVpdp, ............ (vi.)
since
fsdi/V Vp = J
The flux through any closed surface depends merely on Sp.
Comparing (n.) and (vi.) we see that Vp an 4 Sp play a comple-
mentary rdle in these two relations. Various forms may be
found for D t G on which we cannot delay.
Replacing p by or in the second form of the identity
(Art. 126 (iv.)), we obtain the expression
V'tsW dvV , ,
............... (VIL >
applicable throughout a given region, and this exhibits the nature
of the quaternion p of the present article. If there is no
circulation at the boundary, so that we may put sr = VQ (where
Q is a scalar function) in the surface integral, we have on
replacing p by VQ in the identity already referred to
also putting p = ^ in the first form of the same identity and
introducing a new scalar function JR,
- f
Substituting for the surface integral from (vm.) in (vn.) and
attending to the definition of R in (ix.), we find
Moreover R is given by (ix.) as a scalar surface integral depend-
ing on the values of Sdi/or and of Q over the boundary, and
V 2 J?=0 throughout the region. In this notation (n.) and (vi.)
become
(xi.)
If 7 = 0, the distribution of the vectors ET is irrotational ; if P
is zero there is no divergence and the distribution is solenoidal ;
ART. 132.] IRROTATIONAL AND SOLENOIDAL VECTORS. 235
if P and 4 both vanish, the distribution is irrotational and
solenoidal.
If, as in Art. 130 (xvin.), dn is the strength of a tube of vectors
Wzar of cross-section Tdo>, and if dp is along the tube, we have
War . dv= VVcr . TduTdp = dpdn because dp \\ do> 11 VVsr.
If the tubes form closed rings and if di; is the directed element of
a surface bounded by a ring, we find (compare (x.))
, ,-,
F '
or agan
where fi is the solid angle subtended at the extremity of p by
the closed ring of strength dn,
because Sdv / VT(p'-/>)- 1 = Sdy / U( / > / -p). T(p'-p)- s -dQ.
(See Chap. VII, Ex. 22, p. 86.) .
Hence at any point outside the vortex rings, i.e. at a point at
which p does not equal p, we have
- , ....... (XIIL)
This well-known transformation is due to the fact that under
the supposed conditions a certain quaternion is reduced to zero
by the operation of V.
ART. 132. By means of the transformations
r, ...(L)
which may be verified without difficulty, we obtain the trans-
formations,
fsr . dv= f/oSVcr . dv- jpSdi/GJ
Another transformation, likewise depending on the invariantal
properties of V, is
...(m.)
236 THE OPERATOR V. [CHAP. xvi.
and by introducing p and V into any relation it is generally
possible to find a transformation analogous to these.
Ex. 1. The momentum and the moment of momentum with respect to
the origin of vectors p of a portion of a continuous medium of density c,
may be thrown into the forms
).dv~T
and the kinetic energy of the portion may be represented by
^==jcTcr^=-iJcS^
(a) For an incompressible substance of uniform density, if 2= W<r,
A. = c Jcrdv = - c J/)Sdvcr = c J/wdu - i c J/>Vd vcr,
/A = c J V/xrdv = - 2c JpSpedv + c JpSpd v<r = f. c\pVpedv - J c J/o Vp Vdi/cr,
^= $ c jTcAb = - i c JS/ood w + 2c JSpo-edv.
Ex. 2. In the notation of Art. 131, the kinetic energy may be expressed
by
y=-ijc(Si ? cdv4-(P+^)
and for an incompressible substance of uniform density,
T= - f cJ(S^o-dv + RBo-dv)
and the volume integral is
(vii) Equations of motion of a deformable system.
ART. 133. For any system of particles the equations (compare
Arts. 119 and 120, p. 194)
m = / oc .................. (I.)
are independent of the mutual reactions of the particles com-
posing the system, M being the total mass, a- the velocity of the
centre of the mass, r the vector from the centre of mass to the
particle dm, X the resultant force and ju, the resultant couple
referred to the centre of mass.
Suppose the system of particles to compose a definite portion
of a distribution of matter, and let each particle dm be acted
on by a force dm and a couple v\ dm due to external causes.
In addition the portion of matter is subject to the interaction
between it and the rest of the matter. The forces of the
interaction on the portion may be supposed to be the resultant
of a number of forces $dj/ acting at each point of the boundary
ART. 133.] KINETICS OF CONTINUOUS MEDIUM. 237
of the portion, and <frdi/ is a linear function of the tensor of dv
the vector element of the surface. Moreover if c is the density,
we have dm = cdi;, where dv is an element of the volume. The
equation (i.) therefore may be replaced by
(n.)
and D,. fv T T.cdv=fc(,;+VT^)d'y+fvT*di/; ......... (m.)
and the volume integrals are taken throughout the selected
portion while the surface integrals are taken over its boundary.
When we take the portion of matter to be small, the volume
integrals in (n.) are ultimately of the third order of small
quantities and the surface integral is of the second order.
Provided therefore J) t <r is not excessively large for very small
portions and provided $dv is a continuous function of the
vector- element of surface di/, the surface integral must vanish
independently of the volume integrals when the dimensions of
the portion are greatly reduced; and if the portion is taken to be
a tetrahedron whose vector faces are proportional to a, /3, y and
8, we see that the function $di/ at any point must satisfy the
condition
(iv.)
for all vectors a, /3 and y, because we have for the evanescent
tetrahedron $a + 3/3 + $y + $S = 0, where a+/3+y+<5 = 0. Thus
$ is a linear and vector function. We may therefore apply the
integration theorem of Art. 125, Ex. 2, and replace |<3?di/ in (il.)
f
by the volume integral l<I>V.di?, in which V operates on 3?
in situ. Thus we have
v; .................. (v.)
and when we reduce the portion, we find in the limit
Dr- f+c^.SV, ..................... (vi.)
where D^r is the acceleration of the centre of mass of a small
portion of the matter.
Applying the same principles of continuity and of dimensions
to (ill.), and taking the portion of matter to be a small parallele-
piped whose edges are parallel to a, /3 and y, we find
or simply (Art. 67, Ex. 7, p. 97)
............................ (VIL)
238 THE OPERATOE V. [CHAP. xvi.
where e is the spin- vector of <3>, as we see more easily by putting
i, j and k for a, /3 and y. Provided there is no voluminal
distribution of couple, the function <3? is self-conjugate.
The equation of continuity is
c = SV(ccr) or -cS / o iy o 2 /) 3 =C r , ........ ; ...... (vni.)
according as we use Euler's or Lagrange's method (Art. 129), and'
by Art. 128 (vi.) or (xiv.) we may replace (vi.) by
Ex. 1. Find the equation of motion for a perfect fluid.
[The force <3?dv on the boundary of a portion of the fluid is pdv, where p
is the pressure, remembering that dv is outwardly directed. Hence the
equation is Dto-^-'
Ex. 2. Integrating along a stream line, show that
is constant for an element of the matter, and find the integral in the case of
a fluid acted on by conservative forces.
Ex. 3. When the forces acting on a perfect fluid are conservative, the
circulation in any circuit moving with the fluid remains unchanged provided
the density is a function of the pressure.
[We have Dcr= - V(/- > + Jc^dp). See Art. 130, Ex. 1. An independent
proof is easily obtained by Lagrange's method, which gives
and if this vanishes for all closed circuits VVDwr=0.]
Ex. 4. If F= - JSo-dp, show that
AUT. 134. To determine the nature of the stress-function $
for a viscous fluid, we assume as usual that the stress consists of
a hydrostatic pressure and of a part linear in the rate of dis-
tortion of the fluid, and that the stress-function is coaxial with
the strain-function. In the notation of Art. 124, the strain-
function is K0+00> an( i ^ e general linear function coaxial
with this function and linear in its coefficients is of the form
n^+tp'^+rtm", where n and TZ,' are constants and where
m"( = S Vo-) is the first invariant of <j> or <j> or !(0 + 0')-
Consequently the stress-function is of the form
$a=-j9a+^(^ + / )a+^m // a, .................. (l.)
a being an arbitrary vector and p being a hydrostatic pressure.
The hydrostatic pressure is defined more particularly (with
changed sign) to be the mean of the principal stresses, or
. IBS.] VISCOUS FLUID. 239-
Hence the coefficients n and n' are connected by the relation
2?i+3??/=0; ........................... (ii.)
and finally in terms of V (Art. 124, p. 211),
(ill.)
If 7i does not vary from point to point of the fluid, the equation
of motion becomes
; ......... (iv.)
otherwise if n varies, it must undergo operation by the V which
replaces a.
In like manner for an isotropic elastic solid, if 6 is the
displacement,
(v.)
assuming that the stress function is coaxial with the strain-
function and linear in its constituents. The equation of motion
becomes
9 ............. (vi.)
ART. 135. The rate of change of kinetic energy of any finite
portion of the matter is
D f
. dv = D iTo* . dm
and in the last integral V operates on <f> but not on o- as indicated
by the suffix. Because So-3? V = S<r $V + S<r$ V, where V operates
on the unsuffixed symbols, we may integrate by parts, and we find
where di/ is an outwardly directed element of the boundary of
the portion of matter.
For comparison we give the expression for the rate of change
of kinetic energy in any region fixed in space. It is
A f JcTcr 2 . dv = f JcTWv - JcS<7<r . df
on making substitutions from the equations of continuity and of
motion.
240 THE OPERATOR V. [CHAP. xvi.
Now -So- VSo- <r= +S<7 V . JTo- 2 and the first integral changes
^t once into a surface integral so that
- cSof. du+ s<r$ V . dv- Sc7$d v , . . .(ill.)
transformation of the second part of the integral being as before.
The difference between (IL) and (in.) is due to the influx of
matter through the boundary.
The first integral in (IL) is due to the activity of the applied
forces ; the third is due to that of the surface stresses ; the second,
with sign changed, gives the rate at which energy is stored in
the medium or dissipated.
ART. 136. In the case of a viscous fluid, the rate of storage
.and waste of energy per unit volume is (Art. 134 (ill.))
- S<r$ V =pS V<r + %(S VV / Snr'+ S Vo-'S VV) - -MS V<r) 2 . . . .(l.)
By the aid of the equation of continuity (Art. 133 (vm.)) the
term in p may be replaced by
- l db, ............... (n.)
where b is the bulkiness, the reciprocal of the density ; and for a
given mass the rate of change of the intrinsic energy is
( pS Vo- , dv = - JpDt&dm = - D (dm [pdb .......... (in.)
The part quadratic in a- is called by Lord Rayleigh the
dissipation function, and it measures the rate at which energy
per unit volume is wasted by the viscosity. This depends on the
distortion, and it is expressible in terms of the elongations
^, e% and e z the latent roots of the function 0o = H0 + 0') of
Art. 124.
The invariant m' of is (Art. 124 (VIL), p. 213)
m' = - iS V V V 7 Vcro-' = }S V<rS VV - JS Vo-'S W ;
also we have
= S V Vo-V V V = S Vc/S W - S V V'S<7</ ;
from these two expressions we get
SVV / S(7<r / +SVcr / SVV= -4e 2 ~4m / +2m //2
since m" = - S V<r = - S VV. Thus
2 F= n(S WScrv + S Vo-'SV'cr - |S Vcr 2 ) = %n(m"* - 3m' - 3 2 ). (iv.)
<* RT - 18 6.] SOLID MOVING IN FLUID. 241
But (Art. 68, p. 98) the invariants of <p Q are
m" = ei +e2+e B and m'+f^efy+eft+efr
and therefore
^=M(o-a 3 )2 + (e 3 -e 1 ) 2 +(6 1 -e 2 ) 2 } ................ (v.)
_ Hence it follows that if the dissipation function vanishes the
distortion of any element must be a uniform dilatation or con-
traction, for the conditions are
Ex. For a dynamical system consisting of a solid and a fluid, the
momentum and the moment of momentum of the system referred to the
centre of mass of the solid are given by
,o> being the angular velocity of the solid, v the velocity of its centre of mass,
4>< the moment of momentum of the solid, p a vector from the centre of
mass of the solid to an element dm of the fluid which is moving with
velocity <r.
(a) In general (//,, A) is the resultant wrench of the system of impulses
which would generate the motion, and if the motion of the fluid is due to
tthat of the solid, A. and ju. are functions of v and o> ; but if the motion can be
generated by applying the wrench to the solid, it follows from Newton's law
of the composition of velocities that A and /JE, are linear functions of v and CD,
or that (p. 208, Ex. 12)
A. = (
where < ls <f> 2 , < 2 ' and < 3 are four linear vector functions.
(b) The work done in altering v and w to v+dv and <i>+do> is
sand if the dynamical system is conservative, so that d W is the differential of
#, function W of v and o>, the functions < t and < 3 must be self -conjugate and
4><2 must be the conjugate of < 2 .
(c) In the case of a perfect fluid, the velocity generated in this way must
be irrotational, and assuming that cr, as well as A and ft, is a linear function
of v and o>, we must have
where 9 and are vector functions of the vector />.
(<#) In the case of a solid moving in an infinite liquid of uniform density,
or of a solid containing a cavity filled with liquid, the functions and f
W&ust satisfy
V 2 0=0, V 2 f=0
{throughout the liquid. And at the surface of the solid in contact with
the liquid
eo that and f m.us$ satisfy the surface conditions
dv=SdvV.0
J.Q,
42 THE OPERATOK V. [CHAP. xvi.
(e) In this case we may replace the expressions for X and p by
and by the aid of the conditions which 6 and f satisfy, it may be shown that
JdvSa0= JSW . 0Saff . dv, J VpdvSaf = JSVV . Saf . dv,
ffydvSafl- JS W . Sa0' . do, JdvSaf = JS W . 0Saf . dv,
so that the conditions (6) are satisfied. Also the functions < 1: < 2 and < 3
depend on the nature of the solid and on the density of the liquid, and they
are invariably related to the solid.
(/) If the solid is acted on by an applied wrench (77, ) referred to its
centre of mass, the equations of motion, analogous to Baler's equation for a
rigid body, are
the second equation being obtained by expressing that the rate of change of
the moment of momentum (//,+VyA) with respect to a fixed point is equal
to the moment of the applied forces (ri+Vyg) with respect to that point.
(g) When there are no applied forces obtain and interpret the integrals
> 1 v + < 2 o>) = const., S (<^! v + <&j<o) (fa'v + ^3^) = const.,
k 3 a> = const.
When the linear momentum is constantly zero,
and the angular velocity is that of a certain solid moving round a fixed point
under the action of the couple 77.
(i) For a steady motion of translation under no forces Vv<^> 1 v=0 ; and in
general for steady motion when o> does not vanish
where # is a scalar. From this it follows that the axis of the screws of
steady motion are parallel to edges of a sextic cone, and in general to each
edge of the cone corresponds a single screw.
ART. 137. In terms o the displacement 6, the equation for
an elastic solid is (compare Art. 134 (vi.))
(I.)
the velocity cr being Q and $ being a self-conjugate function
because there is no voluminal distribution of couple. The
displacement 9 is a function of the time and the position vector,
and when the strain is small we may neglect the term Sf?V .
in D t *6. We replace, in fact, D t *6 by the second derived of 6
regarded as a function of t alone, that is by Q. Observe that
now V is commutative in order of operation with the result
of differentiating with respect to the time,
ART. isr.] ELASTIC SOLID. 243
^ By Art. 135, the rate at which the forces work in storing and
dissipating energy is the integral
(n.)
taken throughout the body. By Hooke's law, stress is a linear
function of strain. If the strain is multiplied by n, the function
$ is likewise multiplied by n. Suppose the strain to be
gradually increased from zero so that at any stage the strain is
n times the final amount where n is positive and less than unity.
In this case (n.) becomes TF= ~-nn\$QJJ ,dv\ and integrating
between the limits and 1, the total work done in producing
the strain in this particular way is seen to be
W=-i\S6$QV.dv (ni.)
If the work done is a function of the strain and not of the
manner in which it has been produced, the function W is the
energy function a quadratic function of the strain, and the work
done in altering the strain in any arbitrary manner is the
difference of the values of the energy function corresponding to
the final and the initial state.
When the energy function exists we see on comparison of (n.)
and (in.) that in general for any two sets of strain answering to
the displacements 1 and 2 , we have
r i*a^i dv = 1 So2$iV . dv (iv.)
j
In fact the theory is quite analogous to that of the linear function
in the quadratic expression Sp<f>p. If dSp<f>p = %Sdp<pp the
* ,1 if* i JO J ^ __ _ O_ -JL ~ -fvv Oil
.
function <j> must be self -conjugate, and S^pg = S/> 2 0/>i for ^ all
vectors. Conversely, if (iv.) holds good for all pairs of strains,
the energy function exists. g
The quaternion statement of Hooke's law is the function $ is
linear in the constituents of the self -conjugate function
In other words, $ is a linear function of V and of ft which is
unchanged when 6 and V are interchanged, V operating ^n aift*
on 0. Thus if a is an arbitrary vector free from the operation
of V, Hooke's law is contained in the equation
$a = 0(a, V, 0) = 6(a, ft V), ................... (v.)
where is a linear function of a, of V and of 0.
24 4 THE OPERATOR V. [CHAP. xvi.
In case the energy function exists
7 V V 2 , 2 ) = S0 1 0(V 1 ,0 2 , V 2 )
2 , V lf 1 ) = S0 2 0(V 2 , 1? V x ) (vi.)
But we have already shown that $ is self-conjugate, so we may
equate the expressions (vi.) to the new expressions
$1> @2> ^2)
0i, V x ) (vii.)
We may sum up the whole matter in the following statement :
writing for four arbitrary vectors
(a 9 fa y , <$)= -Sa0(/3, y, S), (vm.)
the fact that $ is self -conjugate allows us to interchange the
positions of a and /3 ; Hooke's law permits the interchange of y
and 8; the existence of the energy equation renders the pair
a, j8 interchangeable with the pair y, S.
For any system of mutually rectangular unit vectors, i, j, k,
we obtain from (v.) six self -conjugate vector functions (of a),
0(a, i, i\ 0(a, j, j), 0(a, k, k), 6(a, j, *), 6(a, *, i), Q(a, i,j), (IX.)
with permission to interchange the positions of the second and
third vectors. The thirty-six constituents of these functions are
the thirty-six elastic constants in case the energy function does
not exist. When the energy function does exist, the number of
constants is at once reduced to twenty-one; three of the type
(i, i, i, i)\ six (i i, i, j); three (i, ij,fy three (ij, ij)i three
(j, k } i, i) and three (j, i, k, i\ using the notation indicated in
(VIIL).
To exhibit clearly the meaning of these constants we shall
employ a special notation for the strains. Let Q*=iu+jv + kw
and p=ix -\-jy + kz \ let
"dx 3 } *' Jt 'dy 'dx
Then the stress across a directed area a arising from the strain 8#
is 0(a, i, i)s ilt and that arising from the strain % is 0(a, i,j)8y-
The symbol (ijki) represents the component of the stress across
unit area j parallel to i due to unit strain of the type % ; and
when the energy function exists this is equal to the component
parallel to k of the stress across unit area i due to unit strain of
the type s^.
Ex. 1. Show that the energy function is of the form
8 +2(^w^+i2(v&W+S(^)^*
+2(m)>%H
AET. isr.] STRESS IN TERMS OF STRAIN. 245
Ex. 2. Determine the reduction in the number of the elastic constants
when the substance posseses a plane of symmetry.
(a) If the substance has two mutually rectangular planes of symmetry,
the plane at right angles to both is a plane of symmetry.
[Reflection with respect to a plane of symmetry leaves the elastic
properties unchanged. If k is normal to the plane, the constants whose
symbols involve k an odd number of times must vanish. Thirteen of the
twenty-one constants remain. When the substance has two planes^of
symmetry, at right angles to j and to k, only symbols of the types (uii), (Hjj)
and (ijif) remain, and hence the plane normal to i is also a plane of
symmetry.]
Ex. 3. If the elastic constants referred to i, j, k remain unchanged when
the axes of reference, i andy, are turned through two right angles round k,
the plane perpendicular to k is a plane of symmetry.
[In this case change of i and j into i and j must leave the symbols
unchanged.]
Ex. 4. Determine the conditions that the elastic constants may remain
unchanged when i and J are rotated through a finite angle v round k.
[If a and /3 are the vectors obtained by turning ^ and j through an
arbitrary angle u round is, the functions of ?/, (kkk&\ (kaJcfi), etc., must be
periodic functions of u for the period v or else reduce to constants. These
functions can be expressed as sums of sines and cosines of u, 2u, 3u and 4tt
together with constant terms. Hence the only admissible values of v are
TT, f?r or TT. In every case the symbols involving k three times must
vanish. We have already considered'rotation through two right angles. For
rotation through TT, the symbols linear in k must also vanish, and changing
i and/ into +j and -i respectively must leave all symbols unaltered. Thus
(kkii)=(kkjj\ (H*<0=0, etc., and (iiy)+(jjji)=0, (*M)=(w)- For rotation
through |TT the functions of u independent of k or involving k twice must
reduce to constants. We find in addition to the conditions satisfied for
rotation through one right angle that (iiij)=(jjji)=Q> (MM) =("#)+ 2 (i^;).
Expressing that (acca), (aa/3) are functions of cosSw and sm3w, we get
-fyiii\=(kM)=(kijj\ -(kjjfi = (fafi)=(kjii). For rotation through an
arbitrary angle the symbols linear in k must vanish and the conditions for
v=$ir must hold.]
Ex. 5. When the energy function exists prove the existence of a
self-conjugate function < for which the relation
0(a, ft 7) - 0(ft a, y)= V . <Va . 7
is identically true.
(a) The axes of <, when determinate, form a natural system of lines of
reference, and where a plane of symmetry exists, it is normal to an axis.
[The function on the left is obviously a linear function of Vaft Operating
by S8 we have
(Safty) - (BBay) (/tySa) - (/^a) = - S VytyVajS - SVa/ty V^S,
and as this is a symmetrical function of Va/3 and of V 7 8 the self -con jugate
character of < is established.
For an arbitrary set of mutually rectangular axes, we have
r\ / ! / 7/\ /'v / 'i a *t 7 r f \ - V fj\T.J V &fa(*
t/ It 7 K ) ~~ \J ( ? 1 * J *" ) " VlWv . Iv , CU^.,
whence it follows that if i, j and k are the axes of < the vectors are
completely permutable in Q&j, k\ so *w. ^IWrnB. etc.
eaiy -
that if # is normal to a plane of symmetry it is
6 THE OPERATOR V. [CHAP. xvi.
If e 1? <? 2 , 3 are the latent roots of < we have in terms of the axes
Given the constants referred to axes z',/, K we can on transformation to the
axes of <f> determine whether there are planes of symmetry or not.
In general putting p=zk+ra, where a=icosu+jsmu, we have the
expansion
(pppp)=^()+
and when i,j and k are axes of < we have also
(kaaa) = (Mii) cos 3 ^ H- 3 (kiij) cos% sin u + 3 (kijj) cos -w sin% + (kjjj) sin 3 u
because the letters in a symbol involving i^j and k are completely permutable
for this special set of axes. Hence it follows that a plane z=0 which is a
plane of symmetry of the quartic (pppp) and of the quadric Spc^p is a plane
of elastic symmetry. The coefficients of the powers of cosw and sin^ in
(kkka) and in (Jeaaa) must then vanish, and by the special laws of interchange
every coefficient of odd order in k vanishes.
Suppose now that the plane 8,7/0= or w = is a plane of symmetry. The
coefficients of the powers of z must be functions of cos u alone. Thus
(pppp) ~4*a + 42?rb cos w+6sV(ccos 2u+c')+4s?*(deos3u+d'cQS u)
4- r 4 (e cos 4u + e' cos 2 u + e")
suppose. If the plane uv is also a plane of symmetry, this function must
be independent of the sign when we put u=vw, where w is arbitrary.
Hence &smtf=csm2v=dsm3v=^ / 3inv=esin4y==e'sin2v==0,
and unless the quartic is a surface of revolution, the only admissible values
of v are Jir, \TT and JTT. Hence planes of elastic symmetry must intersect at
angles of 90, 60 or 45 if every plane through their intersection is not a
plane of symmetry. Of course in the second and third cases, the quadric
Sp<p is of revolution. There is no difficulty in writing down the elastic
constants for each case.
Suppose two roots of < to be equal so that there are indeterminate
axes in the plane of i and .;', and that it is required to find a natural system
of lines of reference. We may equate to zero the derived with respect
to u of the first of the coefficients (kkka\ (kkaa)+Z(kaka\ (&aaa), (aaaa)
which does not vanish. Determining u from such an equation we take
icosu+JQinu and j cos u- i sin u along with k as the natural axes of refer-
ence. The case in which <f> reduces to a constant will be considered in the
next example.]
Ex. 6. When the energy function exists,
V 2 . 6(p, p, p)=
is a self-conjugate vector function invariantally related to the elastic
structure.
[The function is invariantal because V 2 is an invariant operator inde-
pendent of any particular choice of i, j and k. If a plane of symmetry exists,
it is a principal plane of this function, because if k is normal to a plane of
symmetry, SiVj and S/< 2 & both vanish, being of odd order in k. Therefore
k is an axis of ^> 2 5 an< i $2 an ^ < of ^the last example have a common axis. *
In terms of the axes i, j and k of the last example, it is easy to see that
- Sty*t' = 32 (waa) + 2 (e 2 + <? 3 ), - S%* = 32 (# aa),
where a stands for i,j and k in the summation.
The axes of this function may be used as natural axes of reference when
the function <f> of the last example reduces to a constant e. In this case for
arbitrary axes, i,j and k are completely permutable in any symbol in which
they all occur, and (jkjk)=e+(jjkk), etc.]
ART. IBS.] ELASTIC SYMMETRY. 247
ART. 138. In the notation of the last Article, the equation of
vibrations of an elastic solid, not acted on by voluminal forces, is
C = 9(V,V,0), ........................... (i.)
where, as we have said, 6 is the second partial derived, with
respect to the time, of 0, which is a function of t and p.
Consider the propagation of a plane wave. If the vector v
represents in magnitude and direction the wave-velocity, the
equation of a wave-front is
u = t-SP, ................................. (n.)
v
for this represents a plane moving at right angles to itself with
velocity v. Over a wave-front, the displacement from the mean
position is, by definition, the same at every point at any given
time. In other words is a function of u and of t. Hence
ou v ou
and generally if /V is a homogeneous function of V of order n,
/*-/ ........... ...........
In particular (i.) becomes for plane wave motion
.
If the wave is of permanent type, 6 involves t only as involved
in u, and if in addition the vibration is harmonic and of
frequency p,
In this case (iv.) becomes
6(Uv, Uv, 0) = c0Ti/ 2 ...................... (vi.)
This shows that for a plane wave propagated in the direction
Uu, the vibration is parallel to an axis of the linear vector
function* 6(Ui/, ITu, a), and that the velocity is the square root
of the quotient of the corresponding latent root by the density.
The solid admits of three plane-polarised waves propagated ^ in
the same direction with different velocities. The wave-velocity
surface is determined by the equation
which is equivalent to the latent cubic of the function
Q(Uv, Uu, a).
*The function 6(Uu, Uu, a) is not one of the functions 9 (a, , i) of the last
Article. The second and third vectors may be interchanged in these expressions,
not the first and second.
248 THE OPEBATOB V. [CHAP. xvi.
When the energy function exists, the linear function
9(Ui/, Ui;, a)
is self-conjugate because we have by the law of interchanges
(Art. 137 (VIIL)), S0e(Ui/, Uw, a) = Sa0(Ui/, TJv, /3). In this case
the vibrations 6 V 2 > 3 for any direction of wave propagation
are mutually rectangular. Moreover, since the function W is
essentially positive, the latent roots of the function are positive
as well as real, and there are therefore three real wave-velocities
TTvTi/y Ui/Ti/j and ~UvTv B in any direction.
When a linear function has indeterminate axes, the \[s function
of 0_ g vanishes where g is the repeated root (Art. 66). The
condition for indeterminate directions of vibration is therefore
where a and ft arbitrary vectors.
This equation admits of a finite number of solutions (v), which
correspond to Hamilton's internal conical refraction. These
vectors terminate at double points on the wave- velocity surface.
The index-surface (MacCullagh) or the surface of 'wave-
slowness (Hamilton) is the inverse
of the wave-velocity surface (vn.), the vector /* being equal
to u' 1 .
The wave-surface, or the surface of ray-velocity, is the envelope
of the plane n
-l, ...................... (x.)
subject to the condition (vn.) or (ix.). That is, the wave-surface
is the reciprocal of the index surface with respect to the unit
sphere /r+l = 0; or it is the envelope of plane wave-fronts in
unit time after passing through the origin ; or it is the wave of
the vibration propagated from the origin in unit time ; or the
vectors p which satisfy its equation represent in magnitude and
direction the ray- velocities.
When the energy function exists a simple and remarkable
expression may be found for the ray-velocity p in terms of p
and 0. The wave-surface may be expressed by elimination
between
The second equation is in full
ART. 139.] i PLANE WAVES IN ELASTIC SOLID. 249
and operating on this by 80 and attending to the law of inter-
changes (Art. 137 (viil.)),
2Sd M e(0, e, AO
and by (XL) this reduces to
Thus every dp is perpendicular to 0(0, 0, JUL) and also to p, so
that 0(0, 0, fji) = xp where # is a scalar. Operating by Sju we
find -# = 8^0(0, 0, M ) = S0eGu, AC, 0) = c0 2 , and therefore
0(U0,U0, M ) = cp ......................... (XIL)
Further, if we operate on this by S/x. and on the first of (XL)
by 80 we recover the relation S/o/x + l = 0; so that all the
relations connecting U0, p. and p are comprised in the two
relations
0)=o0, 0(U0, u0, fi=cp
(viii) Electro-magnetic Theory.
ART. 139. The fundamental circuital laws of the electro-
magnetic field are*
(I.) the circulation ( S?yd/>) of the magnetic force (??) in any
closed circuit is equal to the flux ( JSydi/J of the electric
current (y) through the circuit divided by the velocity of light
(u) in free space ;
(II.) the circulation, with changed sign, (+ Sed/>) of the
electric force (e) in any closed circuit is equal to the flux
f ijsy/li;) of the magnetic current (y,) through the circuit
divided by u.
These laws are symbolized by the relations
and because it is implied that the fluxes of the vectors y and y,
through the circuit are independent of any particular surface
bounded by the circuit (Art. 130 (xv.)),
SVy = 0, SV y/ = ......................... (n.)
* We cannot delay to explain the units employed in this article. Full explana-
tion will be found in the article by H. A. Lorentz on Maxwell's Electromagnetische
Theorit in Bd. V 2 , pp. 63-144, of the EncyUopadie, der mathematischen Wissen-
schqften. These units are but slightly modified from Heaviside's rational^units.
Much use has been made of Lorentz's article and of Heaviside's work in the
preparation of the account of the theory given in the text.
250 THE. OPERATOR V. [CHAP. xvr.
We proceed to define more particularly what is meant by the
electric and magnetic current fluxes and by the electric and
magnetic forces in these laws. The electric current flux through
the circuit consists in general of three parts, the flux (
due to the conduction current (i), the rate of change ( D* S<5dy)
of the electric displacement (<5) through the circuit, and the flux
due to the convection current (ev) where e is the
(
density of electrification* carried through the circuit with
velocity v. In like manner the magnetic current is due to the
rate of change ( Djs/3dv) of the magnetic induction (/3) through
the circuit, to a conduction current (*,) postulated by Heaviside,
but probably non-existent, and to a convection current (e f v,)
where e, is the density of magnetification carried through the
circuit with velocity v r On the whole the integral fluxes are
In the rate of change of the displacement through the circuit we
must take account of the motion of the circuit which we suppose
to move with the velocity cr, varying from point to point. We
have therefore by Art. 129 (ill.), p. 229.
...(iv.)
where =<$-VVV<r<5-<rSVc$, g = /3 - V V V<7/3 - <rS V/3 ...... (v.)
Converting the line integrals in (i.) into surface integrals and
expressing that the relations hold for every possible small circuit
di/, we arrive at the differential equations of circuitation
+e/y/ ) ......... (vi.)
We have not yet explained the meaning of the vectors e and 77.
The total electric and magnetic forces at a point consist of
impressed forces (e^ and ^) together with e and ^. Thus if e* and
vjt are the total forces,
(VII.)
* This is not electrification of the medium. It is due to charges of electricity
carried by moving particles, for example.
ART. wo.] ELECTKO-MAGNETIC EQUATIONS. 251
and -Lorentz' further divides the impressed electric force into a
part ac co-operative with e in producing the conduction current
and a part GM co-operative with e in producing the displacement.
We shall write
t = + ic + id, tJt = JJ + rjic+m ............... (VIII.)
where the suffix i calls to mind that the force is impressed,
c that it relates to conduction current, d to displacement and
b to magnetic induction (/3).
Expressing that the conduction currents are produced by the
forces enumerated, we have
* = $(*+*), i, = *,(* + !/*); ................... (ix.)
and by Ohm's law in the case of isotropic media $ is a scalar
the conductivity and for anisotropic media $ is a linear vector
function. Similarly we suppose the postulated function $,
corresponding to the postulated magnetic conduction current y,
to be a linear vector function.
In like manner, expressing that the displacement (5) and the
induction (/3) are due to the forces mentioned,
S = <f>(e+e id ), /8 = 0,(i7+i;a) ................... (X.)
The phenomena of 7 .. ; " v ' : '*~ shows that <j> and <j> / are not
always linear functions of the forces, but we shall only consider
the important case in which they are linear functions. For
isotropic media, <f> is the (scalar) dielectric constant and <f>, is the
magnetic permeability.
Some little care is necessary in differentiating these expres-
sions when the medium is in motion. Owing to the motion <f>
may change its value at a point fixed in space.
ART. 140. The activity of the impressed electric and magnetic
forces with reference to a small element of the medium of volume
dv is
)dt; > ........................... (I.)
transformation being made by (ix.) and (x.) of the last article.
Transforming again by (vi.) we find
+ eSev + ej&nv, + v S V Ye?) . dv, ........ (ll.)
because we
The electric and magnetic forces evoke mechanical forces, f
per unit volume, and the stress $ 8 dv across the directed element
252 THE OPERATOR V. [CHAP. nn.
cU If the element moves with velocity a- the activity of these
forces on the element is
A 2 dv = -(S^+So^VJJdt;, .................. (m.)
the last term, in which (V) operates on $ and on a- in situ,
being equal to the surface integral So-$ s dy over the element.
The total activity Adv^^ + A^dv ........................ (iv.)
is equal to the rate of transfer of energy to the element.
The term /= -Si*- 1 *-^,*,- 1 !, ....................... (v.)
is by Joule's law the rate of waste of energy per unit volume
owing to the conversion of energy into heat by the resistance.
The terms in this expression for the Joulian waste are analogous
to the dissipation function of a viscous fluid. The term
eSev+efiqv, relates to the convection currents.
The work done in increasing the electric displacement by the
amount dS is
(VI.)
where eta is the total electric force operative in producing the
displacement. (Compare (vin.) and (x.) of the last article.)
Experiments on dielectrics show that an energy function exists,
or in other words the work done is the differential of the function
W= - JSfy- l $= -iSed = -iSe^ew, .......... (VII.)
which represents the energy stored in unit volume of the medium
and due to the electric force. From the existence of this energy
function we infer "that <f> is self -conjugate. A similar result
holds good for the magnetic induction, and the energy due to
this cause is
ff> -JS/fy,- 1 ^ - JS^/3= - JSro^ ....... (vm.)
The energy stored in unit of volume due to electric and mag-
netic forces is the sum of W and W r
When the medium is at rest the total activity is (11.)
Adv = (J+ W + W,-eSev-ej&nv,-uVVeii)dv 9 ...... (IX.)
because in this case S and ft must be replaced by S and J3. We
have accounted for every term except the last. This by a pro-
cess of exclusion represents the rate of radiation of energy from
the small volume. It may be expressed as a surface integral,
(x.)
and this is the total outward flux of the vector uV^, the vector
area dp being outwardly directed as usual. This vector is the
ART. HO.] ENERGY IN ELECTRO-MAGNETIC FIELD. 253
Poynting vector discovered independently by Professor Poynting
and Mr. Oliver Heaviside. It represents in magnitude and
direction the flux of radiated energy.
Granting that the same vector represents the energy flux
when the medium is in motion, and there seems to be no adequate
reason for doubt, the total activity is
), ...(XL)
the last term being the rate of change of the energy stored in
the element dv and due to electric and magnetic causes.
Equating this to the sum (A^A^Av already obtained, we have
v. ....(xn.)
By Art. 129 (in.), p. 229, we find
D,( Wdv) = (D, W - TFSV<r) . dv
where D^- 1 is the result of operating by D on the function -1 .
Further, by (iv.) of the same article,
I = D t S - V V'V<r'($ = D t S - <5S Vo- + S<5V. </,
and therefore
Hence equation (xn.) becomes
The first term on the right may be written
S^d^-^S-S<rV3<5 0-%
where V operates on 0- 1 alone since we have generally
where d* refers to the rate of change at a point fixed in space.
Consider now the term $SS . d t <f>~ 1 $> where d^" 1 is the time
rate of change of <p~ l at the extremity of the vector p
This change depends on the rate of distortion and on the angular
velocity of the anisotropic medium. In other words, it is a
function of the operation of V on cr. Let = iu+jv+kw be the
displacement at the point so that or = iu+ji)+ kw, and let u& etc.,
254 THE OPEKATOE V. [CHAP. xvi.
denote the deriveds of u and w with respect to x, y and z. We
have
suppose, where V operates on o- alone, and where
Introducing this function and an analogous function for the
corresponding magnetic term, and accenting vectors a- operated
on by V, we replace (xin.) by
Now this relation, or identity, is formally true for all velo-
cities o-, and for all distortions and angular velocities (iVVo-);
and by the principle of virtual velocities we equate corresponding
terms of the relation. The symbolical statement of this principle
is that the identity (xv.) remains true when we substitute for
<r, V and cr' any three arbitrary vectors X, JUL and y. Hence
because />i=p 2 if SXp 1 = SXp 2 &> r all vectors X ; and again
, ...(xvii.)
since ^=^2 ^ Sv^/x = Si/0 2 ^t for all vectors /m and j/.
Keplace jj. in this expression by V operating in situ on the
various vectors, and we find
because +V ' W^-jS- ^-1/33 V/S-JVS^- 1 ^, (xvm.)
i
and
ART. HI.] STEESS IN ELECTEO-MAGNETIO FIELD. 255
Thus we find for the mechanical force
S ....... (xix.)
The stress across any small area is determined by (XVIL).
In general the terms in 9 and 0, are small, and we shall
neglect them.
The stress across any small area due to the electric displace-
ment is when we neglect 0,
and if JUL is parallel to <j>~ l S, we have
$ c U</>- 1 3--iU^- 1 5.S(5
while if JUL is perpendicular to 0" 1 ,
- 1 ^ -U>. F.
Thus the stress consists of a tension along the lines 'U<j>- 1 S
and an equal pressure at right angles to these lines, numerically
equal to the electric energy per unit volume. Similar results
hold for the magnetic stress.
ART. 141. When the circuit is at rest, and when there is no
convection current the equations of circuitation become
S + i^uVVrj, /3= -uVVe, ..................... (I.)
when we put ^ = 0. When moreover the medium is at rest we
have (Art. 139 (x.) and (ix.))
<5 = 0(e + e w ), j8 = &(* + *) < = $(e+e ltf ); ............ (IL>
and from these we obtain the equation
00?+ez^ + $<+<^
which is explicit in the vector e. Having determined e from
this equation, the impressed forces being known, we obtain <J, i
and j8 by direct operations on e. The vectors 77 and VV^ are also
expressible by direct operations in terms of e.
There are two principal types of this equation. For a
dielectric non-conductor $ is zero, and the propagation of the
disturbance is by waves. For a conductor incapable of storing
electric energy, is zero and the propagation is by diffusion,
When there are no applied forces the equations (i.) and (11.)
may be replaced by
0e+<e=uVV> 7 , 0,77 + ^77= -u We; ............ (IV.)
and assuming e=Ea n e n e M , r\ = 2a n i7 n e 6 * , ..................... (v.)
256 ,' THE OPERATOE V. [CHAP. rn.
where the a and the b are constant scalars, the equations are
identically satisfied provided e n and r\ n satisfy the equations
&0e n + *6 n = uVVifo, ftn^n + ^n^-uWen ....... (VI.)
and the boundary conditions. The scalars b must in general be
determined by an equation arising from the boundary conditions.
The scalars a depend on the initial state of the disturbance.
The particular solutions n e b ^ r) n e b -^ are the normal solutions, and for
any two normal solutions we have
ttSWerts + & 2 Si4> 2 + ^n^i + Si$ a + &n&fli = 0, ......... (VIL)
because S We^g = S^ 2 VV'e/ - Scj V V^'. Integrating throughout the medium
and converting a volume integral into a surface integral we find,
u JSVc^dv + & 2 jSe^dv + ^ {S^/^dv + {Se^dv + JS^/^dv = ;
wJS V^dv + &! JSe^dv + b 2 JSifc^adv + JScg^dt; + JSTj^/^dv = 0, (vin.)
the second equation following by interchange of suffixes from the first.
If in either of these equations we replace 6 t and & 2 by conjugate complex
expressions b'*J~^lb", and at the same time replace 1 and 2 by 'V~l"
and ^ and 173 by ?/ db \/ - l-i/', the real part of the equations is
ttJS(WV+ v ^dv+&'J(^
+ {(SeW + Ss'W + 817'*^' + ST?"^?/') di; = 0, . . . (ix.)
remembering in the reduction of this expression that < and <j5> y are self-
conjugate (Art. 140 (vii.)). The surface integral is the total inward flux of
energy across the boundary due to the disturbances c', 97' and e", ?/'. If no
energy is communicated from outside the boundary, this is zero or negative
zero if no energy from inside escapes, and otherwise negative. The
remaining integrals are all negative, the coefficient of b' being minus double
the energy stored by the two distributions separately and the remaining
integral being minus the energy wasted by conductive friction. Hence in
any case 6' cannot be positive. If there is no energy radiated and none
dissipated, b' must be zero or else ', e", >?' and 77" must vanish so that there is
no disturbance. On the whole then, the real parts of the scalars b are zero or
negative when the medium receives no external energy ; when in addition
there is no dissipation and no radiation of energy across the boundary the
real parts are zero, and in this case there are permanent oscillations within
the medium, the scalars a being determined once for all by the initial
conditions.
ART. 142. We shall now give a sketch of the theory of the
propagation of light in a crystalline medium adopting Clerk
Maxwell's hypothesis. The medium being supposed non-con-
ducting the functions $ and $, disappear, and the equations of
A free vibration become
............ (I.)
when <j> and 0, are two self-conjugate functions which are
constant if the properties of the medium are the same for the
same directions at all points.*
^*The suffixes are employed in these equations as we shall have more to deal
-with the vectors e ajad y denned in (n.).
ART. 142.] MAXWELL'S THEORY OF LIGHT. 257
Assuming for a plane wave (Art. 138, p. 247) that
........... (n.)
where v is the wave-velocity, we find on substitution in (i.)
5=0 6 =uVu- 1 j;, /3=^=-uVi;- 1 e ............. (III.)
From these we obtain among other relations
-tt?^Se<5 = Se0e = uSev- 1 j 7 =:S>70 / >7 = S>7/3; ......... (IV.)
which show that the magnetic energy per unit volume is equal
to the electric energy, for we have
W, ............. (v.)
The total energy is ^(;sin* 2 7^U S^J, and the mean energy is
consequently %w. v
Again if p represents the ray-velocity we have
S=l, Spdir^O ........................ (vi.)
for all differentials dv. Differentiating (m.)
......... (vn.)
operating by Se on the first, or by Sj; on the second, and
attending to (in.), we find
Sdi/^Vc^O, ........................... (vin.)
because by (iv.) Sed<S and S/3djy are each equal to ^dw.
As this holds for all values of dv we must have p parallel to
*;, and by (iv.) we find for the ray-velocity
and this, it should be noticed, is parallel to the Poynting Flux
(Art. 140). Again it is easy to deduce from (in.) and (iv.) the
expression for the wave- velocity (v)
, uw
=
We have now enumerated six vectors depending on the
propagation of the wave which are connected by the relations,
P = T,^ *I> 5== T^
J.Q.
258 THE OPEEATOR V. [CHAP. xvt.
and these vectors when drawn from a common origin pierce a
concentric sphere in a pair of supplemental triangles^
When some one of the four vectors /3, S, e and ^ is given, all
the vectors can in general be determined subject to a choice of
sign. If e is given, we have <$ = 0e, w = - Se<5 and
for the equations give S*iS=Q, S/3e=0, or
and the suitable tensor is found by substituting tj^xVfafa in
<w= Sq<j>,r}. Hence /3, p and v~ l can be found without ambiguity
when the sign is selected. The case of exception is when e (or rf)
is a solution of the equation
V0a0,a = 0, ........................... (XIIL)
or, in other words, an axis of the (generally non-conjugate)
function 0~V/ or <," V-
When Ui/ or U/> is given, two independent values of the
vectors can in general be found, and the solution corresponds to
the splitting up of a wave or ray into two plane polarised waves
travelling with a given direction for the wave- or the ray- velocity.
Let us seek to determine S and /? from the second and third of
(XI.) when Uu is given. We have
<rt ........... (xiv.)
and from these, when we eliminate /3 and S in turn, and introduce
new linear functions <f> v and <f> jv , we find
,
J ...... ( }
Thus S is an axis of the linear vector function denoted by <
and Tv 2 is the corresponding root, and because <f> v has one zero
root (corresponding to the axis <j>Uv) there are only two finite
latent roots or two values of the wave-velocity along the
direction Uu. That the functions <f> v and <t>^ have the same
latent roots appears from the fact that their latent cubics are
equivalent to the equation in Tu 2 obtained by eliminating /3 and
8 from (xiv.). if Ti/ 2 is the second root of <f> v and if $' is the
corresponding axis, we have
and therefore (by (xiv.)), since Tv 2 is not generally equal to Tv' 2 ,
SJ0-V-0, S/S^-^O ................... (xvi.)
But these conditions may be written in the form
S&^S^S/V == 8/3^=0,
ART. 142.] PLANE POLARISED WAVES. 259
where e' ? ft i/, etc., correspond to S'. Thus & is perpendicular to
e and v, and therefore parallel to ft by (XL), and ft is parallel to
8. In fact we have
'= +U/3, U/3'= TUS, .................. (xvm.)
because Uv - U V<5/3 = UV<S'/3'.
Since S and (S' satisfy the relations (compare (xvi.))
we easily find on putting Uv=UW== W:TW in (xv.) that
u 2 S<r0 / - 1 5 / S<S0- 1 5=T u 2 TVdT 2 , ............... (xx.)
and that
This result leads to a simple construction. Let the quadrics
cr= 1 and 8^,"%= -1 ............ (xxii.)
be constructed. Then by (xix.) S and <T are paraUel to the pair
of common conjugate radii in the central plane at right angles
to the direction of the wave-velocity. Let or and or, be respec-
tively the vector radii of the first and second quadrics parallel
to S, and let 57' and cr/ be those parallel to ^, then we have
U (XXIII.)
and from this construction everything relating to the wave can
be determined. For the first set of signs in (xvili.) we have
(XXIV.)
where i</ is double the mean energy per unit volume for the
second wave.* (Compare (iv.).)
From the fifth and sixth of equations (XL) we have
eTp-^tt^V^Up, ifTp-^u^VUp.^e; ........ (xxv.)
and as in (xv.) we may write,
= - f ,
......... >
that ^-^ has the same direction as the central perpendicular on the
tangent plane to the quadric StD'^>~ 1 ^= - 1 at the extremity of t7 and that its
length is the reciprocal of that of the perpendicular.
260
THE OPEEATOR V.
[CHAP. xvi.
and if we take e and e" to be the two axes of $ p corresponding
to the two finite latent roots T/r 2 and Tp"" 2 of the function, we
find as before
for it appears that W= TLy, W= q: Ue.
We can write down results analogous to (xxm.) and (xxiv.)
for the various vectors related to the waves whose ray- velocity
is along a fixed direction U/>.
"We now return to equation (XIL), which we may write in the form
(xxvii.)
i/
where m t is the third invariant of < y and where *
because we have
Expressing p and v" 1 in terms of , by (XL),
P ~~ fj{m,w(v/w, V?) }~ \/{ m,w (w'w, -w 2 ) }'
From these equations, on attending to (XXVIIL),
"'; (xxxi.)
rL-lAyi -1 = Jff . .(XXXII.)
V ^J{mfiv(w r w f -w^)}' '
' l )p=y?(w,<l)w<pj)v~ 1 ; (xxxin.)
the last relations, which alone are likely to give trouble, being derived from
(xxxu.) by operating with (w f w$~ l $)(yD$~ l <$> w) on both sides, remember-
ing that in this the factors are commutative.
From (xxx.) and (xxxi.) the equations of the wave-velocity surface and of
the ray-velocity surface may be written down, and equations (xxxm.) and
(xxxiv.) are suitable for investigating the cases of indeterminations which
correspond to external and internal conical refraction.
Suppose, for example, that w'=b*w, where 6 2 is a latent root of the function
^x^/" 1 and that /3 (not now the magnetic induction) is the corresponding axis
while ft 1 is the axis of the conjugate function <f>^ 1 cf> corresponding to the same
*It should be noticed that w? has not here its recent meaning.
ART. H2.] ELECTBO-MAGJSTETIC WAYE-SUKFACE. 261
root. The equation (xxxiu.) fails to give a determinate value of p, and
operating on it by S/3', we find
S/3'< v -i=o, ................................. (xxxv.)
since <^>^ / =6 2 ^j8 / . Two other equations for v are obtained by putting
w'b^w in (xxxi.), and these are
- ^r 1 )" 1 V la =0, SirtyiT 1 - -m y &%- 2 ; ......... (xxxvi.)
and from these three equations we find four values of or 1 , say v{~ 1 and
ug" 1 . Substituting the value iT 1 in (xxxiu.) and replacing w t by its value
in terms of p by (xxx.), we get
and this equation represents a plane conic. For we have seen (xxxv.) that
each vector in this expression is perpendicular to /?', so that if of and y are
the remaining axes of <$>r l <$> corresponding to the latent roots a 2 and c 2 , the
equation is equivalent to the pair
In order to calculate in the most explicit manner the vectors *u 1 ~ 1 , etc., we
may by Art. 71, p. 100, reduce the functions <, and < to the trinomial forms
where identically \ = - SaSa'A = - Sa'Sa A.
Putting v~ l = afp + /3'q + 7'r, equation (xxxv.) becomes #=0, while (xxxvi.)
reduces to ^) 2 (c" 2 ~6~ 2 )=^(6- 2 -a- 2 ) and^? 2 +r 2 =m y 6 2 ^- 2 , and we finally get
for the four vectors
Again, taking p=ax+/3y+yz, and substituting in (xxxvu.), we find a
simple expression
2 =0 (XL.)
for the equation of the conic traced out by the extremity of p. We notice
In order to obtain more explicit forms for the equations of the wave-
surface and the wave- velocity surface, we note that the first equation (xxx.)
expands into
' = 0,
where >/r and T/O are Hamilton's auxiliary functions and where
^YA./z=Y^X^+Y^ / X<#>f t -
By the aid of the second equation (xxx.) this becomes
o ......................... (XLI.)
In like manner
o ........... (XLII.)
is the equation of the wave- velocity surface, where
2 THE OPEBATOE V. [CHAP. xvi.
Other forms may be given to the equation of the wave-surface such as
derived from (XLI.), and
derived from (xxx.) by the aid of the trinomial expressions for the functions,
but in problems treated by quaternions it is frequently preferable to deal
directly with vector expressions rather than with the scalar equations of
surfaces obtained by eliminating certain quantities from the vector equations.
Ex. Show that the wave-surface may be derived from a Fresnel's wave-
surface by a pure strain.
[Put p=p' and &p<p&-w$fr l p=8p\wftw-wn)r l p\ also
p'*, eta]
CHAPTER XVII.
PBOJECTIYE GEOMETRY.
ART. 143. There are several interpretations which may be
assigned to a quaternion and which we have not yet explained.
We now propose to show that a quaternion is capable of repre-
senting a definite point loaded with a definite weight or mass,
and throughout this chapter we shall speak rather indifferently
of quaternions or of points.*
In the identity
2 = Sgr.(l+|Z)-Sgr.(l + OQ) if OQ-^, (l.)
it is manifest that the point Q at the extremity of the vector OQ
drawn from an assumed origin is determined when the qua-
ternion q is given, and that Sq is also determined. We regard
Sq as a weight or a mass concentrated at the point. We shall
sometimes use capital letters concurrently with small letters,
? =Q.Sg, Q=1+OQ, (II.)
to denote points of unit weight, or unit points, so that Q.w
denotes the point Q weighted with w. Thus SQ= 1, VQ=OQ.
The difference of two unit points is the vector joining them,
Q-p=l + OQ-(H-OP)=OQ-OP-PQ; (IIL)
and the origin is the scalar point
= 1 (iv.)
A vector represents the point at infinity along its direction, as
appears by allowing Sq to diminish indefinitely in (i.) while Vg
remains constant, for OQ will then increase indefinitely in length,
so that at last V# represents the point at infinity in its direction.
* See Tram. R.LA., vol. xxxii, and Phil. Trans., vol. 201, pt. viii. I regret
that at the time of publication of these papers I was not acquainted with an able
memoir by Dr. James Byrnie Shaw (American Journal of Mathematics, vol. xix.,
pp. 193-216), in which somewhat similar results are obtained.
264 PEOJECTIVE GEOMETBY. [CHAP. xvn.
The relation
.................. (v.)
contains the principle of the centre of mass. It asserts that the
point p+q+r is situated at the centre of mass of p, q and r,
and that its weight S(p + q+r) is the sum of the weights of the
three points. In another form,
l + a s )
Ex, 1. The middle point of the line-AB is
Ex 2. Interpret the relation
regarding Sp+Vg', etc., as representing weighted points.
Ex. 3. The centre of mass of equal and opposite weights is at infinity.
Ex. 4. The equations of the line a, b and of the plane a, b, c are
where #, y and z are scalars.
Ex. 5. Corresponding points of similar divisions on the, lines ab and
cd are _
a , , b c d f
S5 + *SP S5 + *63'
and corresponding points of homographic divisions on the same lines are
a+tb, c+td,
t being a variable scalar.
[See Art. 3*7, p. 41.]
Ex. 6. The equation q=a+2bt+ct* represents a conic.
Ex.7. The equation ... , _.
^ q=a+tb+u(c+td)
represents a ruled quadrie, t and u being variable scalars.
ART. 144. In order to develop this method, it becomes neces-
sary to employ certain special symbols, and with one exception
these are to be found in Art. 365 of Hamilton's Elements of
Quaternions, though in quite a different connection.
For any pair of points, we write
(a, &)=&Sa-aS6, [a, &]=V.VaV&; ............... ( L )
a,nd in particular, for points of unit weight (A = 1 + a , B = 1 + B)
these hecome ^ J>
(A, B)=B-A-- a , [A, B]=V.VAVB = V a /3 = Va(/3-a). (II.)
ART. u*.] SPECIAL SYMBOLS. 265
Thus (a, b) is the product of the weights into the vector connect-
ing the points, and [db] is the product of the weights into the
moment of the vector connecting the points with respect to
the scalar point or origin. The two functions (a, b) and [a, b]
completely determine the line ab.
For any three points we write
[a, 6, c](a, 6, c)-[6,
(a, 6, c)=S[a, 6,c]S
and for unit points A = l + a, B = l+/3, C=l + y, these become
[A, B, C] = Say-V/3y-Vya- Va/3, (ABC)-S . a/3y. ...(IV.)
Hence it appears that the quaternion [a, 6, c] determines the
plane of the points, and regarded as a point symbol [a, 6, c]
represents the reciprocal of the plane with respect to the unit
sphere having its centre < at the scalar point. For the vector
V[a6c] : S[a6c] is minus the reciprocal of the vector perpendicular
from the origin on the plane SpV(/3y-f ya + a/3) = Sa/3y ; that is,
its extremity terminates at the pole of the plane with respect to
the unit sphere. The symbol (a, b, c) is the sextupled volume of
the pyramid OABC multiplied by the weights SaSSSc.
Any quaternion may therefore be regarded as representing at
pleasure a plane or a point reciprocals with respect to the unit
sphere.
The last special symbol we require at present is
(v.)
or for unit points,
(vi.)
Thus (AJBCD) is the sextupled volume of the tetrahedron ABCD,
and (abed) is the same volume multiplied by the product of the
weights.
It will be observed that the five functions are combinatorial,
that is to say, they remain unchanged when to any of the
quaternions involved in one of the functions is added a sum of
products of the other quaternions multiplied by scalar coefficients.
For example, [a+xb+yc, b, c] = [a, 6, c]. More generally when
the constituent quaternions are replaced by linear functions of
themselves with scalar multipliers, the functions are merely
multiplied by a scalar. If any linear relation with scalar co-
efficients connects the constituents of a function, the value of
the function is zero. If any two constituents are transposed the
function changes sign, and in fact the laws of combination of
the rows or columns of an ordinary scalar determinant are
obeyed by the constituents of the functions.
266 PROJECTIYE GEOMETRY. [CHAP. xvn.
ART. 145. In terms o these functions, the equation of the
line db and of the plane abc are respectively
[q, a, 6] = 0, (q, a, 6, c)=0 ; ..................... (i.)
the first expressing that q, a and b are linearly connected, or
that the plane qab is indeterminate ; the second requiring the
volume (QABC) to be zero.
The equation of the line ab may also be written in the form
(pqab)=Q, .............................. (li.)
where p is a point wholly arbitrary ; and the equation of the
plane may be replaced by
SqlQ, where Z=[a6c], .................... (m.)
the point I being, as we have said, the reciprocal of the plane
with respect to the unit sphere*
S.2 2 =0, ..... , ........................ (iv.)
orS.(l+OQ) 2 ==0,orOQ 2 +l==0. Putting L = 1+OL, the equation
of the plane takes the known vector form S(1+OQ)(1+OL) =
The plane at infinity is
S2 = 0, ................................. (v.)
this being the reciprocal of the scalar point (the centre) with
respect to the unit sphere ; or otherwise if q represents a point
at infinity it is a vector (Art. 143, p. 263), so that Sg = 0.
The formulae of reciprocation
([abc]i [abd]) = [ab](abcd); [[o6c]; [abd]]= ~(ab)(abcd), (vi.)
are worthy of notice. They connect two points a and b with
two points [abc] and [abd] on the reciprocal of the line ab, and
are easily verified by vectors. Formulae, such as these, are
often suggested by the forms of the expressions. For example,
the left-hand members of the above relations evidently vanish
if a, b, c and d are linearly connected. We infer that (abed) is a
factor, and the remaining factor must be a combination of (ab)
and [ab}.
It is often useful to observe that if i, j and k are mutually
rectangular unit vectors,
[ij, *]=-!, (1, *, &) -1 ;
and relations such as these may be employed to ascertain the
numerical factors in expressions such as (vi.).
^%^I y * h ? m ?*T^ . coordinates tlie auxiliary quadric is generally taken
to be ^+^++,^=0. It is more convenient in quaternions to employ the
Art 'ft T ^ 1 ^ hOW6Ver n 10S " f 8**- (Compare
ART. us.] PEINCIPLE OF DUALITY. 267
Ex. 1. Two lines, a, &, and c, d t intersect if
(abcd)=0.
(a) This condition may be also written in the form
Ex. 2. The point of intersection of three planes
S^=0, Sw2=0, Snq=0 is <? =[7, m, ri\.
Ex. 3. The line of intersection of two planes SZj 0, Smq=Q is
q=\l,m,n\
where n is an arbitrary quaternion.
Ex. 4. If four planes ?, m, n, p have a common point
Ex. 5. The line , b intersects the plane 8^=0 in the point
- bSla.
Ex 6. The general equation of a conic is
where t is a scalar parameter.
(a) The expression q = atfa +b(^+ t 2 ) + c
represents the pole of the chord joining the points ^ and t%, or the tangent
at ti if t 2 is variable.
(b) The pole of the line in which the plane Slq=Q meets that of the
conic is
(c) The centre is q = aSc - 26S6 + cSa.
(d) The conic is a parabola if SaSc=(S6) 2 .
(e) What kind of a conic is represented by
(/) If q, q-t, q% q z and q are any five points on a conic, and if t, t^ t% # 3
and 4 are the corresponding parameters, the anharmonic of the pencil
Ex. 7. The general twisted cubic is
<?=(, 6, c,
(#) The equation #=(#, 6, c, c$ifi, 1^ I) 2
represents the tangent at the point J? 2 , *i being variable.
(b) The osculating plane at a point is
two of the scalars jfi, ? 2 , * 3 being variable and the other being fixed.
(c) The equation in (a) represents the tangent line developable when
ti and t z both vary.
(d) If 1 is given it represents the conic in which the osculating plane at
L cuts the developable.
(e) The locus of the poles of a fixed plane S^=0 with respect to these
conies is the conic,
268 .PROJECTTVE GEOMETKY. [CHAP. xvn.
(/) The osculating planes at the points in which the plane SlqQ meets
the curve intersect in the point
q
and this point lies in the plane.
(g) The symbol of the osculating plane Bpq0 at the point t is
p=[at+b, bt+c, ct+d] ;
and this equation also represents the cuspidal edge of the reciprocal
developable.
(h) The last equation may be written in the form
(i) The symbol of the plane containing three points ft, 2 ? h ^
(j) The anharmonic of the group of planes joining two variable points on
the cubic to four fixed points is constant.
ART. 146. Hamilton has given two relations connecting five
arbitrary quaternions,
a(bcde)+ b(cdea)+c(deab)+ d(eabc)+e(abcd) = ...... (i.)
and e(abcd)=[bcd]Sae-[acd]Sbe+[abd]8ce-[abc]8de' y ...(n.)
which are of great importance and which correspond to the
vector relations
SSafiy = aS/3yS+pS*yaS+ySa/3S = V/3ySa<5+ VyaS/3(5 + Va/3Sy<$.
The first has been virtually proved in Art. 39, p. 43, and we
may at once verify it by writing
xa+yb+zc+wd+ve=Q,
where x, y, z, w and v are scalars to be determined. From this,
by the combinatorial property, we have
0=(a, b, c, xa+yb+zc+wd+ve) = (a, b, c, wd+ve),
which gives the ratio of w to v. This relation enables us to
express any point in terms of four given points, so that we may
if we choose use an arbitrary tetrahedron of reference, for
example abed.
The second shows how to refer any point to four given planes
and the truth of the formula may be verified by observing that
we get consistent results when we operate with Sa, S6, Sc
and Sd.
It will be observed that the relations (i.) and (n.) are linear
with respect to each of the five quaternions, so that the weights
of the points do not enter. In fact, just as in tetrahedral
coordinates, geometrical relations depend on homogeneous func-
tions of the quaternions. Though it is in general distinctly
disadvantageous to employ any system of coordinates in
ART. 146.] BELATIONS CONNECTING FIVE QUATEENIONS. 269
quaternion investigations, or even to refer in thought to any
tetrahedron or axes of reference until a problem has been
reduced to its ultimate simplicity, yet it is worth while observ-
ing that if we express a variable quaternion q in terms of four
given quaternions a, b, c, d by means of the relation
q=xa+yb+zc+ivd, (m.)
the scalars x t y, z and w are the anharmonic coordinates of
Art. 40, p. 43.
Ex. 1. The line de meets the plane abc in the point
d(abce) e(abcd).
Ex. 2. Show that
([060],
[[abc], [def]] = -(ef)(abcd)-(fd)(abce)-(de)(abcf).
[Compare Art. 145 (vi.). Four points on the line of intersection of the
planes a&c and def are d(abce)-e(abcd) and d(abcf)f(abcd\ and the
functions [a'fc'] and (a'b f ) for two points on the line are proportional to
the right-hand members of the above. The weights are correct, and it only
remains to determine the numerical factors. Putting d=a and e=6, we
verify the signs by the equations cited.]
Ex. 3. The point of intersection of the planes a&c, def and ghi is
b c
Wl
(bdef) (cdef)
(aghi) (bghi) (cghi)
[Equating the left-hand member to xa+yb+sc, we have
x(adef)+y(bdef)+z(cdef)=Q, etc.,
and to determine the factor we may put
The left-hand member becomes +1, and the determinant also reduces
to +1.]
Ex. 4. Given four triangles a n b n c n , where %=1, 2, 3 or 4, show that six
times the volume of the tetrahedron determined by their planes is
(a 1 a 3 6 3 c 3 )
[This follows from the last example.]
Ex. 5. Establish the identities
Sa6' Sac'
S6&' S&c'
Sea' Sc&' Sec'
Saa' Sa6' Sac' Sad'
Sba' Sbb' Sbc f Sbd'
Sea' ScV Sec' Sea"
SaV SdV SaV Sdd'
-S[a6c][a'6'c'];
270 PEOJECTIVE GEOMETKY. [CHAP. xvn.
[The first determinant is combinatorial in #-, b and c and also in a', b'
and c f . It vanishes if either triangle reduces to a line, and conversely.
Hence it must be a scalar function of [abc] and of [a'&V], that is (having
regard to the weights) it must be of the form
where x and y are numerical factors. For a=a'=i, &&'=/, c=c'=k we
get y 1, and for &=a'=l, b=b'ij c=c'=j we find .#= l.j
Ex. 6. Prove that
[This is most easily proved by vectors. Compare Art. 145, Ex. 1.]
Ex. 7. Find the equation of the hyperboloid having three given
generators ab, a'b' and a r b".
[There are various methods of finding this equation, but we shall give a
method to illustrate the use of Ex. 3. If p and q are any two points on
a generator of the opposite system to the given lines, the conditions of
intersection are (pqao)=Q, (pqa'b')=Q, (pqa'b")Q. Kegarding these con-
ditions as the equations of planes, p being the variable point, the condition
that the planes should intersect in a line is [[##&] [tfa'&TV/ "]] ). which
becomes (aqcfV)(l>q<fV) - (bqa'b')(aqa"b") =0.]
ART. 147. The results of the last article are particular cases
of a very general theory applicable not only to quaternions but
to any operators or quantities which are associative and
commutative in addition.*
If /(a, b) is a function of two quaternions distributive with
respect to each, the function
f(a, b)-f(b, a) .......................... (i.)
is combinatorial in a and 6, for it remains unchanged when we
replace a by a+yb or b by b+m, because
f(a+yb, 6)=/(a, 6)+y/(6, b) and /(ft, a+yb)=f(b, a)+yf(b, b).
In like manner if /(a, 6, c) is distributive with respect to a, b
and c the function /(a, 6, c) f(b, a, c) is combinatorial in a and
b i the function formed by subtracting from this the result, of
interchanging a and c is combinatorial in a and b and also in
a and c ; and the function of six terms
S/(a,&,c) .............................. (ir.)
formed by transposing a, b and c in /(a, &, c) in every possible
way, by changing the sign after every transposition of a pair of
constituents and by adding the results together, is combinatorial
in a, b and c. Similarly if /(a, 6, c, d) is distributive in a, 6, c
and d, the sum 2/(a, 6, c, d) ........................... (in.)
*See an interesting paper by Prof. A. S. Hathaway, Proc. Acad. of
Science, 1897.
ART. 147.] COMBINATORIAL FUNCTIONS. 271
is combinatorial in a, b, c and d ; and finally
2/(a, 6, c, d, e) ........................ (iv.)
is, combinatorial in a, b, c, d, e and vanishes identically because
the five quaternions are linearly connected.
It is geometrically evident from Art. 144, that every com-
binatorial function of two quaternions a and b must be a function
of (ab) and [ab~] the two vectors which determine the line ab.
Every combinatorial function of a, b and c must be a function of
[abc] which determines the plane abc ; and the only combinatorial
function of four points is (abed) the sextupled volume of the
tetrahedron determined by them. Hence (n.) is a linear function
of [abc] and (ill.) is the product of a quaternion by the scalar
(abed).
Now in forming these sums, we may proceed step by step.
For example, let us transpose bcde in f(a, b, c, cZ, e), leaving a
/unchanged. We obtain the sum
where, the temporary suffix applied to a denotes that it is free
from the operation indicated by S. Next interchange a and b
and change the sign and permute a, c, d } e, leaving b unchanged.
Finally the vanishing combinatorial function (iv.) is expanded
in the form
2 f(a Q bcde) - S f (\acde) + 2 f(c Q abde) - 2 f(d abce)
and this general result includes Art. 146 (i.) as a particular case.
Again we may leave two or more quaternions fixed and add
together the sums obtained, so that for example
2f(a Q b Q cd)-2f(a c bd)+ete. = 2f(abcd) ....... (vi.)
These expansions correspond to the expansions of determinants
by minors.
Ex. Find the sources of the functions
([abc], d), [[abc], d],
which are combinatorial in a, b and c, or in other words find linear functions
of a, b, c from which the combinatorial functions may be derived by
summation and transposition.
[Since (abc) . Vd= [be] S . aVd+ [ca] S . bVd+ [ab]S . cVd
and V[abc]Sd=- " '^,-r ~ , ">"* ~ ' "* = ".
the first expression is 2 6cSac? . Similarly the second expression is
- V . [be] Vd . Sa - V . [ca] VdSb - V . [ab] WSc,
and the function may be derived from - V6SVcVc? . Sa or f rom -
272 PROJECTIVE GEOMETRY. [CHAP. xvn.
certain parts of this latter expression vanishing under transposition and
summation. As a determinant, the function is
[[abc]d]=
a b a
Sa S& Sc
Sad Sbd Bed
and this may be deduced directly as follows. We may assume
[[abc~\d]=xa+yb+zc since
and we have
The numerical factor of the determinant resulting from this may be
determined by substituting special values for a, 6, c, d.]
ART. 148. We shall now consider the general linear trans-
formation of points in space.
In analogy with the linear vector function, the linear
quaternion f unction fg is a function which satisfies
f(a+b)=fa+fb ........................... (i.)
for all pairs of quaternions a and 6.
The relation P=fy .............................. (ii.)
represents the general linear transformation from points q to
points p, lines and planes
q=a+tb, q a+tb+uc,
becoming lines and planes
p=fa+tfb, q=fa+tfb+ufc,
and anharinonic properties being preserved.
If four given quaternions, a, b, c and d, are converted by a
linear transformation into four others, a!, b', c' and d', the
function which effects this transformation is (compare Art 62
(iv.), p. 88, and Art. 146 (i.)
fq = - {a f (bcdq)+V(cdqa) +c'(dqab)+d'(qabc)} (abcd)~ l ; (in.)
and this function is in the quadrinomial form. To reduce a
function to the quadrinomial form, we may arbitrarily assume
any four quaternions a, 6, c, d and use either of the relations
connecting five quaternions. Taking the second,
/?= {f[bccl]Saq-f[acd]Sbq+f[abd'}Scq
-f[abc]Sdq}(abcd)-\ ..... (iv.)
and thus a linear quaternion function depends on sixteen
constants, four constants being involved in each of the four
quaternions f[bcd], etc.
In (ni.) we supposed the weights given. Let us nOw determine
a function which shall convert five given points A, B, C, D, E into
AKT. 150.] LINEAK TEANSFOE3MATIOJST. 273
five others A", B', C', D', E', paying no attention to the weights.
Such a function is
B / (ACDg)(A / C / D / E / )
(BCDA)(BCDE) (ACDB)(ACDE)
D / (ABCg)(A / B'CflB / ) .
(ABDC)(ABDE) + (ABCD)(ABCE) ; "^ V ''
for replacing q by A we get /A = A / (B / C / D / E / )(BC.DE)- 1 , etc., and
putting # = E, we have /E = E / (A / B / C'D / )(ABCD)- 1 in virtue of the
relation connecting five quaternipns. Thus the function (v.)
efiects the required transformation, and it is evidently deter-
minate to a scalar factor. (Compare Art. 65, Ex. 5, p. 92.)
ART. 149. A linear function f being regarded as producing
a transformation of points, the inverse of its conjugate f' 1
produces the corresponding tangential transformation.
For any quaternions p and q,
Spq = Spf-*q' = Sq'f-*p = Sq'p, if q'=fq, p,=f-*p ....... (t)
Hence any plane Spq = 0, in which q is the current point and
p the symbol of the plane, becomes after the transformation
829,3' = 0, where q' is the transformed current point and where p,
is the transformed symbol of the plane. In other words when
points are transformed by the operation of /, planes are trans-
formed by the operation of f~ l .
ART. 150. Now the symbol of the plane may be expressed in
terms of three points in the plane (Art. 145, p. 266), and therefore
for some scalar factor n }
nf-^abo] = [fa,fb,fc]=F'[a, 6, c], ............ (i.)
since we may either transform the symbol of the plane in one
step by/ 7 " 1 or we may transform the points a, 6, c which enter
into the symbol by /. The function F' is a new linear function
analogous to Hamilton's \f/ 9 and it is connected with f' 1 by the
relation %
The scalar n may be explicitly expressed in terms of four
arbitrary points, a, b, c, d, by operating with S .fd on (L), when
we find n(abcd) = (fafbfcfd) = S[abc]Ffd, ............ (m.)
where F is the conjugate of F'.
Thus in addition to (n.) we have,
n=fF=Ff; ........................... (iv.)
and we may also write
n(abcd)=(fafbfcfd)= (f'afbfcf'd), \
F[abc] = [f'afbfc]=nf-i[abc]. J
J.Q. S
.
'
274 PEOJECTIYE GEOMETRY. [CHAP. xvn.
Replacing/ by f+t, where t is a scalar, the relations
%=7i+to'+*V'+*V"+^ (vi.)
are obtained, where the new scalars ri, n", n'" and the new
linear functions G and H are defined by
ri(abcd) = 2(afbfcfd) ; n"(abcd) = 2
= [/a, 6,
Moreover, on account of the arbitrariness of t in (vi.),
n=fF, ri=fG+F, n"=fH+G, n'"=f+H-, ...(vm.)
and from the symbolical equations may be deduced the following
explicit expressions for the auxiliary functions
H=n'"-f; Q^ n "- n "'f+^- F=n'-n"f+<nf "/*-/*; ( Ix .)
and the symbolic quartic
satisfied by the function /.
ART. 151. Let t l9 t^ t s and t^ be the roots of the scalar quartic
#~-n"'1?+ri't*-n't+n=Q, .................... (i.)
so that the symbolic quartic may be expressed in the form
(/yW-W-W-O^O ................... (ii.)
It follows just as in the case of the vector function that
(Mi)?i = 0, where (/ 8 )(/-< B )(/-* 4 )? = ?p ...... (m)
and that q l is a fixed point a united point of the transformation
one of four q l9 g 2 , q z and g 4 . The point q is quite arbitrary.
The equations
.
represent respectively a united plane of the transformation and
a united line the plane [? 2 , g s , grj and the line q z q
We have also by the property of the conjugate,
Sg 1 > = Sg i y-i5 1 )g = if (y-^ft'-O; .......... (v.)
and thus the united points ( ?1 ', q z ', q^ and g/) of the conjugate
(f) are the reciprocals with respect to the unit sphere (Art. 145)
of the united planes of /. In other words, the united points of
a Junction and of its conjugate form tetrahedra reciprocal
with respect to the unit sphere.
Ex. 1. Prove that/0 may be reduced to the form
^=(+e)Sg + ST0 + V0,
and determine its latent quartic in terms of the linear vector function <fc
the vectors c and and the scalar e. r
ART. 152.] UNITED POINTS. 275
[By the distributive principle fq-fBq+fVq, etc. To determine the
quartic assume fq==tq=t(Sq + \q), and equate scalar and vector parts. We
find (e-t)Sq +Se / V^=:0, (^-fyVq+tSq^O, so that
Ex. 2. Construct a function with four zero latent roots.
[Assume /a=6, /6=c, /c=c, /c?=0.]
Ex. 3. Examine the nature of the symbolic equation satisfied by the
function a Q cdq } + b(cdqa) + c'(^a&) + d'(qdbc).
[Every point a+ub on the line a, 6, is a united point of the function, and
the JP function of fq-(abcd)q vanishes identically. The quartic degrades
into a cubic.]
Ex. 4. Construct a function satisfying a symbolic quadratic.
[This may arise from one of two causes. The function may have two line
loci of united points a, b and c, d ; or it may have a plane locus of united
points a, 6, c. In the first case the latent quartic is a perfect square. In
the second it has a triple root. For full details on these matters see Phil.
Trans., vol. 201, viii.]
Ex. 5. Prove that two real lines remain unaltered by the general real
linear transformation.
[If the roots are all real of course the six edges of the united tetrahedron
remain unaltered. If the roots are all imaginary, they occur in conjugate
pairs, and the united points must be of the form a*J 16, c \/ Id. The
lines ab and cd are real and remain unchanged.]
ART. 152. Just as in the case of the vector function, we
obtain two new functions
on combining a function and its conjugate by addition and
subtraction.
The function / is self -conjugate and the function f f is the
negative of its conjugate, or
/o=/o'. /,= -// ..... ; ...................... <*)
as we see at once by the property of the conjugate.
Since fq is the general linear function of q, Sqfq or Sqf Q q
is the general scalar quadratic function, and
represents the general quadric surface, the surface being quite
arbitrary both in shape and position, and not now referred to
its centre as in Art. 72, p. 106.
In like manner Sp/# = .............................. C 1 ^.) 1
is the general equation of a linear complex, or of a family of
lines p, q satisfying a single condition of the first order. For if
we replace p by p + tq the equation remains unchanged, for we.
have generally, by the property of the conjugate (n.),
276 PROJECTIYE GEOMETRY. [CHAP. xvn.
The equations Sg/ a = 0, Sqfjb^O ........................ (v.)
represent respectively the polar plane of the point a with respect
to the quadric, and the plane containing the lines of the complex
which pass through 6. The first equation may be deduced from
the result of substituting a+tq in the equation of the quadric,
when we find
Saf Q a + 2tSqf Q a + t*Sqf,q = 0,
and if q is on the polar plane, the points in which the line aq
meets the quadric must be expressible by a+tq, a tq, because
the polar plane is the locus of harmonic means, and the points
a, a+tq, q, atq form a harmonic range.
If 8>lq = Q is an arbitrary plane we see on comparison with (v.)
that the pole of the plane with respect to the quadric is f ~ l l,
and that the point of concourse of the lines of the complex which
lie in the plane is f,~ l l. It also appears that
O and Smf- l l=Q .................. (vi.)
represent respectively the tangential equation of the quadric,
or the equation of the reciprocal quadric; and the tangential
equation of the complex (the intersection of the planes Slq = 0,
Swg = being a line of the complex), or the equation of the
reciprocal complex.
A complete account of the nature of the united points of the
functions / and /, is furnished by the theorem of Art. 151. Since
/ is its own conjugate, each of its united points is reciprocal to
the plane containing the remaining three, or the tetrahedron of
united points is self-conjugate to the sphere of reciprocation.
We^saw in Art. 67, p. 96, that it is impossible for a real self-
conjugate linear vector function to have a pair of equal roots
without having indeterminate axes, and this because a real line
cannot be perpendicular to itself. But a real self-conjugate
linear quaternion function may have two of its united points
coalesced into a single point provided the point is on the sphere
of reciprocation. The argument about real roots does not now
apply. For suppose a+ +/^-lb and a x/--T& to be two united
points of a self-conjugate quaternion function, the condition of
^reciprocity is
and this condition can be satisfied for real points a and 6 if one
point (a) is inside and the other (6) is outside the sphere of
reciprocation Sq* Q.
As regards the function / the most general form its symbolic
quartic can have is
=o or a 2 -*)c/; 2 --*')=o, ......... (vii.)
ART. 152.] QUADEIC SURFACE AND LINEAR COMPLEX. 277
because the same quartic is satisfied by the function and by
its conjugate (/). Supposing the united points to be a, a',
b and V, where
it is evident that a is the united point of the conjugate which
corresponds to the root +/H, etc., and therefore by the theorem
of Art. 151 we must have
=0, Sa&'=0, S& 2 = 0, S&a'=0, 866' = 0,
In other words the lines aa', aft', a'& and &&' are generators of
the unit sphere, or aafW is a quadrilateral on the sphere. The
four lines are consequently all imaginary. By Ex. 5 of the last
article it appears that the lines ab and a'&' must be real ; and
since these lines are reciprocal to the unit sphere, one of them
(ab) meets the sphere in two real points (a and &) and the other
meets it in two imaginary points (of and b*). Consequently one
of the scalars (a) is positive and the other ($') is negative.
The common self-conjugate tetrahedron of two quadrics
Sg/iff 0, Sg/ 2 g = has the united points of f^^ for its vertices.
For if Slq = is the polar of a point a for both quadrics
/ 1 a=^ 1 / 2 a=J or f^f^^a, ............... (vm.)
so that a is a united point and ^ the corresponding latent root of
A~ l fv If & is a second united point corresponding to the root 2>
Sft^a = tjSbfta = Safjb = ^ 2 Sa/ 2 & = 0,
because the functions are self-conjugate. These relations are>
however, geometrical consequences of (vm.) and analogous
expressions.
A little care is necessary when dealing with the equations of
quadrics such as
82.^^3=0 or S ? (/ 1+a; / 2 )(/ 1 + 2 // 2 )-ig=0 ;
the second form of the equation shows that the function involved
is not self -conjugate, although / x and / 2 are self-conjugate, unless
/! is commutative with / 2 .
Ex. 1. In terms of vectors prove that the forms of /and/, are
/o(l+p)=e+ +S />+< p; /(I +/>) = e y - S^+VTJP;
e being a scalar, e , 7 , TJ being vectors and < being a self -con jugate linear
vector function.
Ex. 2. Prove that the latent quartic of the function / is
fi+tW-ifi-frpy-o
and verify the conclusions respecting the roots and united points of/.
278 PKOJECTIVE GEOMETEY. [CHAP. xvn.
Ex. 3. Prove that
is the locus of the poles of tangent planes of the quadric Bqf^^Q with
respect to the quadric Sg/ 2 g=0.
Ex. 4. The locus of the points of concourse of lines of the complex
/#=0 which lie in the tangent planes of the quadric S/j#=0 is the
qu
Ex. 5. An arbitrary quadric and an arbitrary linear complex have a
common quadrilateral of generators.
[This follows by expressing that the point of contact of a plane S^=0
with the quadric <?/(#= is the same as the point of concourse of the lines
of the linear complex &pf,q=Q in the plane. We have /a =tf a=ul, where t
and u are scalars, so that / ~ 1 /^= ta. There are thus four points (a) through
which pairs of the common generators pass, and these points are the united
points of /o-i/J
Ex. 6. If/j and/g are anv two functions, prove that the latent quartics
of /!/2 and of / 2 /] are identical.
(a) Show also that the latent quartic of / ~ */ is of the form
[The first part follows exactly as in the case of vector functions (Art. 71) ;
the second is obtained by combining this principle with the fact that
-//(f * is the conjugate of /(f 1 /,.]
Ex. 7. If a, 6, a' and b' are the united points of the function /o" 1 /,
corresponding to the latent roots -f t, - , + tf -tf, prove that if we take*
_xa+yb , za'+wV tfa+y'b z'a'+w'b'
the equations of the quadric and the linear complex take the canonical forms
Ex. 8. Prove that in any linear transformation the locus of a point
which with its derived is in perspective with a fixed point is a twisted cubic.
[If a is a fixed point, the condition requires [fa, q, a]=0, so that 0, fa
and a are in a line. This equation may be replaced by (f+t)q=ua, or
<?=M(/+*) l a ; and this curve meets an arbitrary plane S^=0 in the three
points determined by the cubic S?(/+2f)~ 1 a=0, or
Ex.9. Prove that (?>/&p,/>)=0
represents the ^quadratic complex of lines connecting points and their
correspondents in the linear transformation produced by/.
(a) Prove that the reciprocal of this complex is "the complex of the
conjugate/, fe/*fc/W-0.
[If p and q are any two points on a line joining a point to its corre-
spondent, we have for some scalars #, y, z, w, the relation xp+yg=f(isp+wq)
The complex foUows on the elimination of the scalars.
If S^=0 and 8^=0 are any two planes through q and its correspondent
Jq, we have S/fy=0, S/m^=0, and for some scalars xl+ym=f(zl+wm).]
Ex. 10. The lines joining points to their correspondents which meet an
Arbitrary right line a, b generate a quadric
ART. 152.] EXAMPLES ON QUADJRIC AND COMPLEX. 279
Ex. 11. An arbitrary quadric S$/ ^=0 has eight generators which join
points to their correspondents in an arbitrary linear transformation.
[If the line , fq is a generator of the quadrie, the point q is one of the
eight intersections of the three surfaces
We shall see that this is the extension of Hamilton's theory of the
" umbilicar generatrices."]
Ex. 12. The generalized normal at a point on a surface being defined as
the line joining the point to the reciprocal of the tangent plane, prove that
the normals of the doubly infinite family of quadrics
compose the quadratic complex (qfqpfp)=0*
Ex. 13. The feet of the generalized normals of the doubly infinite
family which pass through a given point a are given by
where y and z are scalar parameters.
[Any point on the normal to the quadric #, y at the point q may be
written in the form
' where "+*=!> ux+ty=s.]
Ex. 14. The locus of the feet of normals of the family of quadrics
y const, which pass through a given point is a twisted cubic.
Ex. 15. A quadric has eight generators which are also normals.
[Expressing that q=fa+xa is a generator of the quadric &qfq= 0, we
have Sa/a=0, So/%=0, Sa/ 3 a=0, which give eight points a and eight
corresponding normals. See Ex. 11.]
Ex. 16. Find the locus of poles of a fixed plane with respect to the
system of quadrics j- , _,
(a) Prove that the plane Slq=Q touches one quadric if x is fixed, three
if y is fixed, and that if no restriction is placed on x or ?/, the locus of the
points of contact is a conic section.
[Compare generally Exs. 12, 13, 14. In general, if p is a point of contact,
=#Z with the condition Sl.tHl=Q, or
^ f+x f+x
-il, or ^
(since we need not attend to the weight of p). This reduces to a quadratic
in * p
and the locus of p is a conic.]
Ex. 17. The tetrahedron formed by a point and by the poles of the
tangent planes at the point to the three quadrics of a system inscribed in a
developable taken with respect to any fourth quadric of the system, is self-
conjugate with respect to this fourth quadric.
280 PEOJECTIVE GEOMETRY. [CHAP. xvn.
[The equation of a system of quadrics inscribed in a developable is
^(/i+tf/a)"" 1 ? 8 ^, this beiag th reciprocal of a system passing through a
common curve. If #, y, z are the parameters of the quadrics which pass
through a point p, and if Sg/f^^O is the fourth quadric, the poles of the
tangent planes weMA+xf^p, A(fi+2/A)~ l P> A(A+ Z A)"^ But
and this vanishes since p lies on the three quadrics #, y, z. This in particular
gives the theorem that confocal quadrics cut at right angles.]
Ex. 18. The locus of the poles of a plane Sga=0 to the same system of
quadrics is the line
the locus of the poles of the system of planes Sq(a+tb)0 is the ruled
quadric
?=(/i+#/ 2 )(a+^) or (<lMJ>M(qf$M^
and the locus of the points of contact of the system of planes is the twisted
cubic, 0/ 1 ( a +0)S(a+<6)/ 8 (a + ^
[In reducing the scalar equation of the quadric observe that the
quaternion equation is of the form q=a l -^xa^+t(b l -\-xb^ and apply the
identity Art. 146 (i.) to eliminate the arbitrary weight of "q and the scalars
$ and t.]
Ex. 19. Prove that two planes can be drawn through an arbitrary line
to be conjugate to every quadric of the system.
[If the planes S?(a-{-zf&)=0, Sqfa+tfy^Q are conjugate to the quadrics
S^/ 1 ~ 1 g'=0 and ^qf^~ l q=^ the conditions of conjugation
lead on elimination of t or tf to a quadratic in t which determines the two
planes in question. The case of exception arises when the line is a generator
of some quadric. The two conditions become equivalent.]
Ex. 20. Examine the particular cases of the twisted cubic locus of
Ex. 18.
[When the line of intersection of the planes is a generator of one of the
quadrics,/! suppose, the locus becomes q=fi(a+ib). This shows that the
points of contact are homographic with the tangent planes Bq(a+tb)0.
When the line of intersection of the planes is not a generator of some
quadric, let S0a=0 and Sqb=Q be the specially selected planes of the last
example, and let Sa(/ l +^/ 2 )a=0, S&(/ : +v/ 2 )&=0 so that u and v are the
parameters of the quadrics touched by the two planes, then the equation of
the cubic becomes
The cubic is plane if (/ 1 a/ 2 a/ 1 6/ 2 6)=0. (See Ex. 9.)
The cubic degrades into a conic if (/j+^J^O, or (^H-^a^O, that is,
if either of the planes is a united plane of / 2 ~ 1 / 1 .]
Ex. 21. Determine the quadrics of the system S^(/ 1 +^/ 2 )~ 1 ^=0 touched
by an arbitrary line.
ART. 152.] QUADBICS INSCRIBED TO DEVELOPABLE. 281
[Taking the line to be the intersection of the planes 8^=0, Sqb=to
of the last example, the condition of contact is most simply obtained by
expressing that the reciprocal line q=a-\-tb touches the reciprocal quadric
Sq(fi+xf 2 )q=Q. Thus we find
or simply (# - u) (x - v) = 0,
so that the line touches the quadrics touched by the planes.]
Ex. 22. Show that the equation of the tangent cone from the extremity
of the vector p to the quadric
. - ^ t
may be written in the form
ST^+tffl^T^O, where
in the notation of Ex. 1, n being equal to 1 or 2.
[The condition that the line of intersection of the planes Saq=Q, S6#=0
should touch the quadric Sg/~ 1 g-=0 may by the last example be written
in the form
(6 + 2S a + Sa^a)(e + 2S ^ + S/5^)
where a=H-a,6=l+/3. This reduces to
and if the line of intersection of the planes is parallel to T and if p is the
vector to a point on it, we may take Ya/3=r, /3-a=-V/3T (see p. 40,
Ex. 4), or /3 a= - YpVa/3. Substituting this last expression for /3 a,
we find that the condition becomes
- V p(SeaSp/3 -
or SV a/?V (<a - eSpa - pSea + epSpa) (<j>/3 - cSp/? - pSe/2 + epSp/3) = 0.
In this transformation we make use of the fact that
in order to have the function in the last expression self -con jugate. If then
6X. = $^.- eS A.p - pS \e
the condition becomes 8x^-^=0, and putting /=/i+#/ 2 , and therefore
the result required is obtained.]
Ex. 23. If p is any point ; p l9 p& p s the reciprocals of the tangent planes
to the three confocals (parameters If,, # 2 , t 3 ) which pass through the point ;
show that the tangent cone to any other confocal (parameter t) is
where any point q is expressed in the
[The condition that the line p+uq should touch the confocal t is
Sfi'(/-tO- 1 fl'Sp(/+0- 1 2)-(Sgr(/+0-^) B =0, or
if h is the linear function defined by
Substituting in turn p, p l (=(/+ ^ 1 )~ 1 p), p& and p z for q, we find
282 PKOJECTTVE GEOMETEY. [CHAP. XVIL
because we have
;K/+O-^=(/+O- I (/+O"MP^
which reduces to
x. . n e equaon o e ang
adrics S . q 2 = 0, Sg/<7 = 0.
[If is the point of contact of a tangent line pq to the common curve,
e four conditions S . j9 2 =0, Sp0=0, Sp#=0, Sp/=0, show that
since
The equation 8^=0 reduces to the required form since Spijp2=> etc -l
Ex. 24. Find the equation of the tangent line developable of the
quadrics S . q 2 = 0, Sg/<7 = 0.
[If is the poin
the four conditions
(p,qJpJti=> or
where # and y are two scalars. Substituting for p in the^ conditions of
contact, we find four relations in #, x and ?/, which are easily seen to be
equivalent to three. The second condition gives A =&q(f+x)~ l (f+y)q=Q ;
and because the first and third combine into Sp(f+y)p~fy they give
S<?(/+#)- 2 (/+y)?=0, or
ox
Again the second and fourth give
Sp(/+tf)?=0, or =
To eliminate x and y we have therefore to equate to zero the discriminant of
A with respect to x and to employ the condition .#=0. On expansion A
becomes $q(F+*G+a?ff+a?)(f+y)q-0 9
and as 5=0, this reduces to the quadratic
Sq(F+xO+^E)(
and the discriminant equated to zero gives
Putting for y its value in terms of q the required equation is obtained.]
PY. 25. A plane is drawn through the line a&, and through the line cd
the plane is town which is conjugate to this with respect to the quadric
Sqfq0. The locus of the intersection of the plane is
[If q is a point on the intersection, [qab*] and [qcd] are the symbols of the
two planes. The equation may be transformed by Ex. 5, Art. 146.]
ART. 153. A linear quaternion function has in general sixteen
square roots quite analogous to the square roots of a linear
vector function. A function and its square roots have the same
united points, and the latent roots of the derived functions are
the square roots of those of the original, there being sixteen
different sets according to the choice ol signs. (Compare p. 99.)
In analogy with the reduction of a linear vector function to
the product of a conical rotator and of a self -conjugate function,
we may write
fP=f.f*P, f'P=ft'f*P, where /,=// and /<//=!, ....(i.)
ART. 153.] SQUAEE BOOTS OF LINEAR FUNCTIONS. 283
since if we take/, to be a square root of the product ^ we must
have f t ff = 1 because
JB = JJ = /*/i/</sJ
and thus we have
It appears on counting the constants that f t is not a conical
rotator, there being sixteen constants in / and only ten in the
self-conjugate function / w so that there must be six in f t Con-
sidered geometrically the function f t converts the unit sphere
into itself and leaves unchanged conditions of conjugation with
respect to that sphere, because
S/A/J& = if Sa& = 0.
Further, because ft ft" 1 transformation of symbols of planes
effected by the function f t is identical with that of points
(Art. 149).
To study the nature of a function f t which satisfies the relation
/,//=!=///, or /,=//-* or //=/ r i, ............ (m.)
we shall endeavour to reduce the function to the form
/,=//* where /.=/.', / r = r( )r~ 1 , ............ (iv.)
that is to the product of a self -conjugate function and a rotator.
First we notice that if a function / which satisfies the condition
/r/ r '=l, converts a scalar into a scalar, it is a conical rotator,
affected it may be with a minus sign. For if
/ r
we have for all vectors p,
_
Thus f r p is a vector, and the mutual inclinations of vectors and
their lengths remain unchanged after operation by / r because
8f rP f r 'p=Spp'.
To effect the reduction (iv.), we notice that we must have
because /,//-/,/,///, =/, 2 and f t (l )=//,(!)
Let us now for the sake of symmetry introduce two quaternions
a and b defined by the relations
These quaternions are known when the function f t is given,
and in virtue of (v.),
/ M a = a, /&=-&, Sab-0, ................. (vn.)
so that a and 6 are united points of the function /.
284 PEOJECTIYE GEOMETRY. [CHAP. xvn.
Take any point c conjugate to the line ab, so that Sac = 0,
S6c = ; and take the point d conjugate to the plane abc so that
Sda=0, Sd& = 0, Sd!c = 0. Then we may assume
/c=c, f u d= -d, ....................... (vm.)
and it is evident that all conditions (Art. 152, p. 276) are satisfied
for the self -conjugation of the 'function f u , and that f u *p=p,
where p is any point whatever. The function f u is determined
by the four conditions (VIL) and (vm.), and the rotator f r is
given by f u ~ l f t or by its equivalent f u f t It will be noticed that
there is an infinite number of ways in which this reduction may
be made, for the point G may be any point whatever on the
reciprocal of the line ab. Also the function / has two line loci
of united pointst he line ac and the reciprocal line bd.
Thus we can in an infinite variety of ways reduce an arbitrary
function / to the form
/-/././ ^ere /.-(//)* / M 2 =l, / r =r( )r-i ..... (rt)
As a simple example, consider the transformations which
convert one quadric into another, or which change
Sg/i2 = into Sp/ a p = 0, where p=fq ............ (x.)
We have
/!=///, whence 1 =/,'/, if /=/r*/./i*. ........ ()
and the function f t is quite arbitrary subject to the condition
///,= !
As another example we propose to show that the intersection
of two quadrics is expressible in the form
? = (/"+<)*, ........................... (xii.)
where / is a linear function, t a parameter and a a constant
quaternion.
If this curve lies on the quadric Sg/^ = 0, the relation
must be identically satisfied for all values of t. Now
as appears by squaring both members of each equation, so that
the condition may be written
Sa/Af^
This becomes rational in t if the square roots involving t are
identical, that is if
iWiV/r* or //,=/,/ or if /-/-% ..... (xiv.)
where / 2 is a self -conjugate function, the condition now becoming
=0, or Saa = and
ART. 158.] VAEIOUS TRANSFOEMATIONS. 285
Finally,
S r = (/r 1 /j+*)*a, where Saj^a-O, Sa/ 2 a=0, Sa/^-y^-O (xv.)
is the curve of intersection of the two quadrics Sg/jg^O,
s 2/2? = 9' becailse if we P ut /ry 2 =/2~ Va/i" 1 /* and n tice that
/2./1/2 * s a self -conjugate function, the conditions that the curve
should lie on the second quadric are seen to be the second and
third of the conditions (xv.). Thus a is one of the points of
intersection of three known quadrics.
Ex. 1. Investigate the transformation of one quadric into another by
first transforming to the unit sphere, then transforming the sphere into
itself, and finally transforming the sphere into the second quadric.
[If 83/^=0, 83/22=0 are the two quadrics, the steps are
^ ^, n ?l=/l^ ?2=/tfl, 23=/2~^2,
where /<//=!.]
Ex. 2. Under what conditions can a function /be formed so that for all
points q and q' we shall have
fq and p' =fq' ?
(a) Find the function / when the conditions are satisfied.
[We must have /'/i/=/ 2 with the implied relation //i'/=/ 2 ' connecting
the conjugates of these functions. Hence
W-Ffi-y- 1 -ffif-f* ./r'/i' /,
and therefore the latent roots of the function f 2 ~ 1 fz must be identical with
those of f{~ l f. For if a 2 , 6 2 , C 2j ^2 are ^he united points of /a" 1 /^ and if
t l9 tfr t 3 and # 4 are the corresponding latent roots we have (see p. 100)
frVi-f^^if^^-
Further if #, ^, z and w are certain scalars and if o^, 6 1 , c 1? c^ are the
united points of Ji/i'? we must ^ ave f a 2=$ a i, /^s = ^n f c z ZG D /c?2=^i *
and because ff\f~f^ we have W 2 w 1 =7i 2 , where ?z, w x and % 2 are the fourth
invariants of/,/! and / 2 . But
and subject to this condition #, y, 2 and w are arbitrary, and the function /
involves these arbitrary constants and is given by
fq . (aj
Ex. 3. Under what conditions can two quadrics &qfiq= 0, Sqf z q=Ql>Q
transformed into two others SqfsqOj Sqf 4 q=Q1
[This is nearly the same as the last example. We must have //i/=w/ 3 ,
/'/2/=v/ 4 , where u and v are scalars, and he>ncQf" 1 f 2 ' 1 f l f=uv-' l f^ 1 f St so that
the latent roots of f%" l A and of ff l f must be proportional. In the same
way we obtain the conditions that a linear complex and a quadric should be
simultaneously converted into a linear complex and a quadric.]
Ex. 4. A twisted cubic q=(abcd$t, I) 3 may be converted into another
q f ^(afb'cfd'fa, I) 3 with arbitrary correspondence of points.
f Assuming ^= u , v n where u. v, u' and v' are arbitrary scalars, we
L-cxoocuuxxig yft + tf 9 ' J J
establish a homography connecting the points on one cubic with those on
the other, and if we equate corresponding powers of t in the relation
we have four relations which determine the function/.]
286 PKOJECTIVE GEOMETEY. [CHAP. xvn.
Ex. 5. Prove that
}.*> where s* 2
represents a confocal of a generalized system when two of the parameters
x y, z vary : the intersection of two confocals when only one parameter
varies ; and a point common to the three confocals corresponding to given
values of the parameters. (See p. 124.)
Ex. 6. The generalized confocals are inscribed to the developable of
which ?-(/+*)*
is the cuspidal edge.
[The line of the developable corresponding to x is q-(f+u)(f+x)-e ;
the osculating plane is y (/+)(/+)(/+*)"* 5 the symbol of this plane
/(/+*)"*, /*(/+*)"*! or (/+*)* k A /H or sim P lv
This plane touches every confocal.]
Ex. 7. Eight generators of the circumscribing developable are generators
of an arbitrary quadric of the confocal system.
[The line (f+u)(f+xrfte is a generator of Sg'(/+^)~ 1 ^=0, and this is^one
of eight corresponding to the eight values of e deduced from the conditions
of Ex. 5.]
Ex. 8. Eight rays of the complex of lines joining points to their
correspondents in an arbitrary linear transformation are generators of an
arbitrary quadric.
[The equation of a ray of the complex is g=(/+w)a, where a is arbitrary.
This is a generator of the quadric 8^=0 if So/ia=0, Sa(/ / / 1 +/i/) a== 0>
So/ f / 1 /a=0. This is the generalization of Hamilton's theory of the umbilical
generatrices.]
Ex. 9. The reciprocal of the developable generated by the tangents to
the curve
q=(f+t) m a is p=(f+t)*- m b, where b=*\a>faf*a\
and where m is a given scalar.
Ex. 10. The family of curves q=(f+t) m a includes tie right line, the
conic, the twisted cubic, the quartic intersection of two quadrics, the ex-
cubo quartic and the cuspidal edge of the developable circumscribed to two
quadrics ; the corresponding values of m are 1, 2, - 1 or +3, J, 4 and f.
Ex. 11. The centres of generalized curvature at a point on the quadric
v -i^=0 are
where y and z are the parameters of the confocals which pass through the
point q.
[The point c=(f+u)(f+x)~ 1 ^ is situated on the generalized normal at q
(Ex. 12, p. 279), and if this point remains stationary, that is if it is the
point of intersection of consecutive normals,
since as c is stationary do and c must represent the same point so that
dccdv, where dv is some small scalar. This condition may be replaced by
dq=(f+u)- l (f+w)qdv, where w is a scalar, and operating by &(f+x)~ l q) we
find almost exactly as in Art 82, Ex. 4, p. 122, the required result.]
ART. 153.] GENERALIZED CONFOCALS. 287
Ex. 12. The surface of centres of the quadric x is represented by
Ex. 13. The differential equation of right lines on the surface
where n(y) is the fourth invariant of
[The differential of q=*J{(f+x)(f+y)(f+z)}e is
and the differential equation of right lines on the surface is obtained by
equating to zero
^} Z '(f+^(f+^
Now Se(f+y)- 1 e=n(y)- l &e(F+y@+y2IZ+y 3 ')e=n(y)- l SeFe in virtue of
the conditions satisfied by eJ\
Ex. 14. The differential equation of generalized geodesies on the
surface is
i i
. \-**-0,
where w is a constant of integration.
[A generalized geodesic is a curve whose osculating plane contains the
pole of the tangent plane with respect to the quadric of reciprocation
(S.? 2 0). Thus ((/-l-tf)" 1 ^ q, dq, d 2 q)Q is the differential equation of a
geodesic in terms of q and of its deriveds.
Writing this equation in the form (f+x^q+tq+udq+vdtq^Q, where
, u and v are scalars, operating by Sq, Sdq, S^+tf)" 1 ? and S(/+#)- 1 d#, and
observing that Sdq(f+x)~ 1 dq + Sq(f+x)^ l d 2 q=0^ we deduce
S . g 2 Sdgd a g~ SqdqSqtfq
q S . ? a S . d? 2 - "
This immediately integrates, and we find
s?(/+#)- W?(/+#)~ ld ?==*
where 5 is a scalar constant. By the last example we have
and similarly
; Sqdq^Q,
S . d 2 =i(^-y)(^-^). w(y)- 1 . d^/
Collecting these results "and putting x+sn(x)=w, the required equation
is obtained.]
288 PEOJECTIYE GEOMETRY. [CHAP, xvn
ART. 154. We shall now give a few examples relating to in-
variants of linear transformation and of quadric surfaces, and
shall explain their geometrical import.*
By Art. 150 (v.), p. 273, the relation
(CM)*, (f-W, (f-t)c, (f-t)d)
is an identity for all scalars t and all quaternions a, &, c and d.
In this sense n, n', n" and ri" are invariants, and every
relation connecting them implies some peculiarity in the nature
of the transformation effected by /. But there is a wider sense
in which these four scalars are invariants. If n and n% are the
fourth invariants of two arbitrary functions / x and / 2 , the
relation
- ".
is evidently true since (/$>,/&/& M^n^pqrs), where p, q, T
and s are any quaternions. Thus any relation implying a
peculiarity of the function / and depending on its four scalar
invariants, implies also a corresponding peculiarity in the mutual
relations of the functions /!// 2 and f^ 9 that is, in the relations
of any pair of functions that can be reduced to the forms f-Jf%
and/J 2 . (See p. 98 and Ex. 3, p. 101.)
Ex. 1. If the function / transforms any tetrahedron abed into another
a'b'c'd' having its vertices on the faces of the original, the invariant ri"
vanishes and an infinite number of tetrahedra possess the property. The
converse is also true.
[The conditions are (afbcd)=Q, (ab'cd)=Q, (a&c'd)=0, (abcd')=Q, and
because a'=/a, etc., we find on addition that '/&'"= 0. Let a, b and c be any
arbitrary points, and let d be determined from the first three conditions.
Then we have ri"(abcd)=(abcd'), so that if 7i' ; '=0, the point fd will lie on
the face abo. More generally when ^'"=0 there exists an infinite number
of tetrahedra so that the tetrahedra derived from any one by the operation
of the f unctions /!// 2 and/j/g are related in the manner described.
If ^'=0, the faces of the derived tetrahedra contain the vertices of the
original.]
Ex. 2. The invariant ri" 2 - %n" vanishes whenever a tetrahedron abed is
so related to its correspondent in the transformation, that the tetrahedron
transformed from the correspondent has its vertices on the original.
[The sum of the squares of the latent roots of / is zero, or the first
invariant of/ 2 vanishes.]
Ex. 3. When the invariant
(" /a -4") 8 --64
vanishes it is possible to determine an infinite number of tetrahedra (abed)
*See Phil. Trans., vol. 201, " Quaternions and Protective Geometry."
ART. iM.] INVABIANTS. 289
and their deriveds (a'b'c'd 1 ) so that a tetrahedron can be inscribed to abed
and circumscribed to a'b'c'd'.
[The sum of the square roots of the latent roots of / is zero, or the first
invariant of one of its square roots /* vanishes.]
Ex. 4. If an infinite number of tetrahedra can be inscribed to one
quadric surface and circumscribed to another, find the invariant relation.
[Let abed be the four vertices of a tetrahedron inscribed to the quadric
82/^=0, and let the faces touch S/ 2 <?=0 at tne points a'b'c'd'. If a'=jfo,
etc., we have four equations of inscription So/ia=0, etc. ; twelve equations of
conjugation, Sa'/ 2 6=0, S&'/ 2 a=^0, etc., or Sa// 2 6==0, Sa/ 2 /&=0, etc.; and four
equations of contact Sa'/ 2 a'=0, or Sa// 2 /a==0. The equations of conjugation
require / 2 / to be self -conjugate, or /2//'/2 ; and the equations of contact
/^a^O,
,
may therefore be replaced by Sa/^a^O, etc. Hence if the first invariant
of /is zero and if /2/=/'/ 2 > ^ i g possible to inscribe in the quadric Sqf^qQ
and to circumscribe to Sqfzq=Q an infinite number of tetrahedra. For when
we assume two of the vertices a and 6, we have to determine c and d to satisfy
O, 6, C , <*)=0, ( ? ,/6, c, rf) = 0, (a, b,fy rf)=0, Sc/ 2 f c=0 and Sdf 2 f*d=0.
The first three give d in terms of <?, and on substitution in the fifth we have
two equations in 0, any solution of which will be applicable.
The quadrics Sqftf 2 q=0 and Sqf 2 q=0 possess therefore ;the required
property, and so do the quadrics So'/ 1 ^=0 and 80/^=0, if it is possible to
find a function / for which / a /*=A, /2/=//2 an( * ^'"=0. It is easy to see
that the conditions are satisfied if 'the invariant of the last example vanishes
for the function f^A-]
Ex. 5. If a tetrahedron circumscribed to Sqf^q=Q is self -conjugate to
fyq 9 the first invariant of the f unction f%~ l f$ vanishes.
"[This is virtually proved in the last example, the f unction / 3 being f^f.]
Ex. 6. When the invariant n" vanishes, it is possible to determine an
infinite number of tetrahedra (abed) and their deriveds (a'b'c'd'), so that each
edge (ab) of one of the tetrahedra intersects the opposite edge (c'd') of the
correspondent.
[The invariant is (abcd)n"=^(abc'd'\ and it manifestly vanishes if opposite
edges intersect, that is if each of the six terms (abc'd') vanishes. Conversely
if "=0, we may arbitrarily assume two of the points a and b. We have then
to determine c and d to satisfy five conditions, (abc'd') =0, etc. Solving for d
(Art. 146, Ex. 3, p. 269) from three of these and substituting in the remaining
two, we get two equations quartic in c, and the point c lies on part of the
curve of intersection of the quartic surfaces represented by these equations.]
Ex. 7. Find the locus of intersection of generators of a quadric which
are the sides of a triangle self -con jugate to another quadric.
[If the quadrics are 83/1?= 0, Sg/ 2 =0, we may first reduce the second
quadric to the sphere S# 2 =0 and the first to S#/<7=0 where /=/ 2 "" /i/ ? "
If q is the intersection of the generators and a and b the remaining vertices
of the triangle, the conditions are
Now for the first invariant of /we have
and the conditions require (fa, b, ?,/<?)=0 and (ajb, ?,/?)=0, because the
four constituents of the first are reciprocal to <x, while those of the second are
reciprocal to 6. Also [a, 6, q] =#/, and therefore the locus is
J.Q.
290 PROJECTIVE GEOMETRY. [CHAP. xvn.
Ex. 8. Three intersecting edges of a tetrahedron self-conjugate to one
quadric touch another. Find the locus of the intersection.
[If the tetrahedron qabc is self -con jugate to S^ 2 =0, we have wq=[abc],
xa=[bcql yb=[caq], zc=[abq]; and if the line qa touches 8^=0, the
relation SqfqSafa-Sqfa*=Q must be satisfied. This condition of contact
may be written'in the 'form Sqfq(fa, b, c, q)-$fqa(fq, b, c, #)=0, and there
are two similar conditions of contact obtained from this by cyclical inter-
change of a, b and c. "Writing down the identical relation connecting
a, b, c, q and/<?, and utilizing the conditions, we find
S/?{/?(a, &, c, q)-q(a, 6, c,fq)}-Sqfq{n'"(a, 5, c, q)-(a, b, c,/?)}=0 ;
and this reduces to Sqf 2 q-ri"Sqfq = 0, when the factor S.q* is discarded,
remembering that [abc\=- wq.]
Ex. 9. Each of three planes S0a=0, Sqb=Q, Sqc=0, mutually conjugate
to S^ 2 =0, touches one of the family of confocals Sq(f+u)~ l q=Q. Find the
locus of the intersection of the planes.
[The points 0, a, 6, c satisfy the conditions of the last example which do
not depend on the function /. The conditions of contact are of the form
or
and hence (u -f v + w + ri") S^ 2 - Bqfq =
is the locus required.]
Ex. 10. The edge ab of a tetrahedron self -con jugate to S^ 2 =0 touches
the quadric 8^/^=0. The condition of contact may be reducedTto
and the invariant n" vanishes if all the edges touch the quadric.
[By Ex. 6, Art. 146 and (VL), and Ex. 1, Art. 145, this follows without
trouble.]
Ex. 11. If the functions / 1? / 2 , / 3 , etc., are transformed by multiplying
them by an arbitrary function f x and into an arbitrary function / tf , the
functions fif^-f^ /i/2~ 1 /3/4~ 1 /5j etc., undergo the same transformation and
may be said to be covariant with the original functions for this type of
transformation.
(a) The f unction / m , defined as the coefficient of tfat z in the identity
where t 19 t^ ^ 3 , etc., are arbitrary scalars, is (to a scalar factor) covariant
with the original functions.
(b) Examine the nature of the transformations the inverse and the
conjugate functions undergo simultaneously with the original functions, and
find the condition that self -conjugate properties may be preserved.
ART. 155. Several important geometrical and numerical
relations may be deduced from the identity
0, ...... (I.)
in which p n is a rational and integral homogeneous quaternion
function of q of order m n .
ART. 155.] NUMEEICAL CHAEACTEEISTICS. 291
The scalar equations
represent two surfaces of orders S^m^^ m 4 and ^, 1 5 m l m 6
respectively, and any point on their intersection satisfies the
quaternion equation
or else the three scalar equations
Hence we see that the curve of intersection of the surfaces (n.)
breaks up into two parts, one of which is represented by (ill.),
while the other the complementary curve is common to the
five surfaces (n.) and (iv.).
Now the order of the curve (in.) must be a symmetric function
of m lt m 2 and m s , and that of the complementary curve must be
a symmetric function of the five orders m n . The sum of the
orders is equal to the product of the orders of the surfaces (n.),
that is, to
(S l 5 m l m 4 ) (2 1 5 m 1 m 5 ) S^m^ + S^m^ + S^m^ ;
and accordingly the order of the curve (in.) and that of the
complementary curve are respectively
m 123 = 2 1 3 m 1 2 +S 1 3 m 1 m 2 and m c =S 1 6 7>z 1 m 2 .......... (v.)
Again the points common to the three surfaces (iv.) must
either lie on the surfaces (n.) or else must satisfy the equation
which requires ^ 4 = ^p 5 , where u is a scalar. In the former case
the points lie on the complementary curve. When three surfaces
have no common curve the number of their points of intersection
is the product of their order ; when they have a common curve,
that curve counts for a definite number of points of intersection,
and there are in general other points of intersection not on the
curve.* Now the surfaces (iv.), if they had no common curve,
would intersect in
- m 1 )(2 1 5 m 1 -
m 4 + m 5 ) Z^
common points, the number being transformed so as to exhibit it
as a function of symmetric functions of the five orders and of
symmetric functions of 7?z 4 and m^ The number of points
satisfying (vi.) must be a symmetric function of m 4 and m 5
* Salmon, Three Dimensions, Art. 355.
292 PEOJECTIYE GEOMETBY, [CHAP. xvn.
alone. The number of points of intersection of the surfaces (iv.)
absorbed by the complementary curve is (Three Dimensions,
Art. 355) a linear function of the order and rank of the curve
and the order and rank must both be symmetric functions of
the five orders. Hence the number of solutions of (vi.) is
In the next place, in order to find the rank and the number of
apparent double points of the curve (in.), we notice that it meets
the surface (p 4c p^p- L P^) = ^ in m 123 (2 1 5 'm 1 -m 3 ) points. These
points, as appears from (i.), are either solutions of (piPz) = Q or
points on the complementary curve. The number of intersections
of (m.) with the complementary curve is therefore by (vn.)
h- m *)- t u
S^m^ 3 'E l B m l 2 m 2 m-jn^m^ (vm.)
Employing the relation r+ = m(^+v 2) of Salmon's Three
Dimensions, Art. 346, connecting the rank r and the number of
intersections t of a curve of order m and its complementary on
two surfaces of orders jm and v, we find for the surfaces (n.) of
orders S^m^ m 4 and 'Z l 5 m I m 6 that the rank of the curve
which reduces by (vm.) to
X - 2
s 32 1 3 m 1 S 1 3 m 1 m 2
(ix.)
In the next place, to find the number (A^g) of apparent double
points of the curve (in.), we have (Three Dimensions, Art. 346),
l)~Jr 123 .................... (x.)
The rank (r c } of the complementary curve is determined by
and this may be reduced to
r c = I, l 5 m l 2 l 5 m 1 m 2 +I ll 5 m l m2m 3 -- 22 1 6 m 1 m 2 , ........ (XL)
and the number of apparent double points is
We may denote the complementary curve by the symbol
which is intended to denote that the points of the curve satisfy
every equation obtained by omitting one symbol. Similarly,
Q ................... (xill.)
ART. 156.] ORDER AND BANK OF CURVES. 293
may ^ be taken, to denote the points which satisfy the surfaces
obtained by omitting two symbols. These points lie on the
curve (xu.) and also on the surface Cpip 2< p 3 p 6 ) = 0. But the
intersection of the curve and the surface includes the points t m
on. the curve [pip 2 p B ] = Q. Omitting these, the number of
points is m c (m 1 +m 2 +m 3 + 771^-^23=2^7^2^3 (xiv.)
Ex^ 1. The curve [<?,/#, aj=0, where /is a linear function, is a cubic ; its
rank is 4 and the number of its apparent double points is 1.
Ex. 2. The curve [/i^/^/a^^O is a sextic of rank 16 and with 7
apparent double points. It is the locus of points that can be destroyed by
functions of the system ^1/1 + ^2/2+^3/3) an d the locus of united points of
functions of the system
where t and u are scalars.
Ex.3. The surface Oifr/rf,/rf,/rf)-0
is the locus of united points of a family of linear functions.
(a) When the functions are self-conjugate, it is the Jacobian of four
quadrics.
Ex.4. The curve ((Aq,f&ftf,f&f&q))=Q
is of the tenth order and its rank is 40.
(a) The Jacobians of sets of four out of five given quadrics have a common
curve, and the Jacobians of sets of four out of six quadrics have twenty
common points.
ART. 156. If Q is any homogeneous and scalar function of q
of order m, but not necessarily rational or integral, the equation
Q=o .................................... (i.)
represents a surface.
We shall write the differential of the function Q in the form
........................... (n.)
where p is a homogeneous function of q of order w 1. By
Euler's theorem concerning homogeneous functions, we see by
that =P, ........................... (m.)
where P is the function of p into which Q transforms when q
expressed as a fraction of p is substituted in Q, for we may
regard q as a function of p since p is a determinate function of q.
Again we shall write generally for the differential of p 3
....................... (iv.)
where f q dq is a linear function of dq and where the constituents
of f q involve q in the order m 2; and by Euler's theorem we
have = v.
294 PROJECTIVE GEOMETRY. [CHAP. xvn.
This f unction f q is self -conjugate, as we have shown in a more
general case (Art. 60 (iv.), p. 79).
Now if we differentiate (ill.) we have
d$ = Spdq+Sqdp = dP, ..................... (vi.)
and on comparison with (11.) we see that
dP = nSqdp, where (n l)(m !)==!, ............ (vn.)
and it is easy to verify that n is the order in which p is involved
mp.
' We shall also write generally for the differential of q expressed
as a function of p, dq = (n-I)f p dp, ........................ (via)
and the function f p is also self-conjugate and involves p in the
order n 2 in its constitution. Thus for any differential by (iv.)
and (VIIL) we have
Ap = (m-l)f& = (m-l)(n-I)fJ p .&p**f g f p .dp ...(ix.)
by (vn.)> and accordingly
or one function produces on an arbitrary quaternion the same
effect as the reciprocal of the other. In particular, applying
Euler's theorem to (viu.) as we have already applied it to (iv.),
we obtain the relations
When dq instead of being perfectly arbitrary satisfies
dQ = 0, or Spdq = Q where Q = 0, ............ (xn.)
dq represents some point in the tangent plane at q, and p is the
symbol of the tangent plane or the reciprocal of the plane with
respect to the auxiliary quadric. The equation P=0 is that
of the reciprocal of the surface. The relations of reciprocity are
clearly exhibited by the equations (compare (a), (in.) and (vi.))
= 0, Sqdp=Q, <LP=0, P = if dQ = 0, Q=0;(xm.)
, d 2 P = if also d 2 Q = ...... (xiv.)
Consecutive tangent planes at q and q+dq intersect in the
line common to the planes
&pr=Q, Sdpr = 0, ......................... (xv.)
r beiug the current point, and if q+d'q is a consecutive point on
this edge we have the group of relations
'g = 0, Sdpd'g^O,
= 0, ....... (xvi.)
ART. 156.] THE GENERAL SURFACE. 295
remembering that in general Sdpd'q = Sd'pdq because f q is self-
conjugate. Hence to conjugate* tangents (qdq and qd'q) on the
surface correspond conjugate tangents on the reciprocal, and the
reciprocal of a tangent to the surface is the correspondent of
the conjugate tangent, for we have S(p+xd'p)(q+ydq) = Q.
The differential equation of the asymptotic lines is
0, ............................ (XVIL).
these lines being their own conjugates.
The differential equation of lines of curvature is
(pqdpdq) = Q, ........................ (xvin.)
for this is the condition that consecutive generalized normals
should intersect. If c is a centre of curvature, we have
c=q+tp, dc^(l+tf q )dq+pdt^(q+tp)du y ...... (xix.)
where di6 and dt are some small scalars. (Compare Art. 153,
Ex. 11.) Hence as #>=/<$ we obtain the relation
and operating by 8f q q we get
S?/^- 1 + *)- 1 ?-0 OT 8q(f q -*+t)-*q = 0, ....... (XX.)
since
The theory of generalized curvature is thus connected with
that of the generalized confocals. The scalar t is the parameter
of one of the confocals Sr(f p +t)- l r=Q which pass through g,
r being the current variable. The confocal = is Srf q r = Q.
The roots of this equation in t determine the centres of cur-
vature, and because in terms of f p (~f q ~ l ) it becomes
Sq(F p +tG p +t*H 9 +P)q = Q or Sq(G p +t*H p +t*)q = Q (XXL)
(since F p = n p f p " l = n p f q and Sqf q q 0) after discarding the
factor t, it reduces to a quadratic and gives two values of t.
Ex. 1. The points having common polar planes with respect to two
surfaces satisfy the equation
the points having collinear polar planes with respect to three surfaces lie on
thecurve
the points having concurrent, polar planes with respect to four surfaces
generate the Jacobian (frptf&j = ;
the points having concurrent polar planes with respect to five surfaces lie on
thecurve
* Consecutive tangent planes intersect in the tangent line conjugate to that
joining their points of contact.
296 PBOJECTIYE GEOMETBY. [CHAP. xvn.
and the points having concurrent polar planes with respect to six surfaces
satisfy the equation
provided we write generally <3LQ n =m n Sp n &q, where =0 is the equation of
one of the surfaces.
Ex. 2. To find the osculating plane at a point on the curve of intersection
of two given surfaces.
[The osculating plane must pass through the intersection of the tangent
planes at the point #, and its equation must be of the form
where Sp^O and 8^=0 are the tangent planes. We have identically
Sptf = Spa = Sftdg = Sp 2 d? 0,
and by (xiv.) the scalar t is determined by the condition
Sd^dg'-KSdp.jd? =0,
so that the osculating plane is
This has now to be simplified. Assuming a quaternion a satisfying
Badq = 0, we have dq [pip 2 a J- ^ so &Pi ( m i " l)/!^ dp% ( m -2 - l)A&q, and
accordingly
since f\abe\=[f- l aFbf- l c\. By Art. 146, Ex. 5, this becomes
Saq
Hence the osculating plane is
ART. 157. If we use the notation d a to denote that the
differential of q is equal to a quaternion a, we shall have for the
h ih polar of a with respect to the surface Q = 0,
d/Q = where da=sO, ...................... (i.)
and if m is the order of the surface, we may write the equation
of the & tb polar in the form
d/d r -*Q = ............................. (ir.)
the quaternion r being now the variable point, and r being
regarded as constant in performing the differentiations indicated
If we write (,.)
we may consider the quaternion p to be derived from the scalar Q
by an operator D analogous to Hamilton's operator V, and we
shall have generally and symbolically,
ART. 157.] VARIOUS NOTATIONS. 297
and in particular when q-w+ix+jy+Jcz, a = l, b = i, c=j and
d=k, we have
In this notation (i.) and (11.) become
(SaD)*.Q = 0, (SoD)*(SrD)-*.Q = ............ (VL)
We may also formally identify our notation with Aronhold's
symbolic notation by writing the second of these expressions in
the forms (Soe)*(Sre)-*=0 or e a V>-*=0, ............ (vn.)
where e is a symbolic quaternion devoid of meaning unless it
enters into a term homogeneous in e and of order m, and where
e r =Ser.
There is thus a considerable latitude in the choice of an
appropriate notation for the investigation of project! ve properties
of curves and surfaces.
Ex. 1. In investigations which involve differentials of the third order of
the equation of an arbitrary surface of order m, we may write
d =mSpa, d d d a = m(m- l)S&/a, d c d 6 d a =wi(m-l)(?7i-2)Sc/ 2 (a,6)
with liberty to transpose in any way the quaternions a, 6, c, the function
/ 2 (a, b) being a bilinear function of a and b (compare Art. 60).
(a) In terms of the operator D,
(b) We may also write
where e is a symbolic quaternion devoid of meaning unless it occurs thrice in
a term.
(c) We have
And when we differentiate fa totally we find
d ./a=/. da+(w-2)/ 2 (a, dq).
(d) The equation of the Hessian is
w = or (fa,fb,fc,fd)=0,
where n is the fourth invariant of jf and where a, 6, c and d are arbitrary
points. It may also be expressed in the forms
where e, ', e", e'", etc., are equivalent symbols (compare Art. 147, p. 270).
298 PBOJECTIYE GEOMETRY. [CHAP. xvn.
Ex. 2. If J=(piP2P3Pj) an< ^ dCn^^wSpwdy, where Q n is homogeneous
and of order m n in the variable q,
. J= [toptpd ft - LPlftPJ % + l>lftP J 3 - [ WaiPs] 4-
(a) For an arbitrary differential, and for an arbitrary scalar m,
(b) If four surfaces have a common point, their Jacobian passes through
that point. If the orders of the surfaces are all equal the point of common
intersection is double on the Jacobian. If the orders of three of the surfaces
are equal, the fourth touches the Jacobian. If the orders of two surfaces are
equal, the line of inter-section of the third and fourth touches the Jacobian.
(c) At a point common to the intersection of four surfaces of the same
order m,
2.dV=-w(m-l)2[2? 2 p 3 pJSd/ 1 d2', where dp n =(wi-])/ n dg r ;
and hence the equation of the tangent cone at the double point is
where a is an arbitrary constant quaternion.
(d) If four surfaces have a common multiple-point of order &, we find that
where 2 and 2 ' denote sums of terms which vanish when q coincides with
the multiple point, and we also have
d*! = wijSd^d* -^ -I- vanishing terras.
(e) At the multiple point d 4 *- 5 J and d 4 *' 4 . qJ vanish, and therefore d 4 *- 4 */
vanishes (as in (6)), and the Jacobian has a multiple point of order 4-3 ;
and because we may write (as in (a))
it follows when the surfaces are all of the same order that the Jacobian has
a multiple point of order 4&-2.
Ex. 3. Determine the equation of a surface which meets a given surface
at the points of contact of lines which meet it in four consecutive points.
[This investigation, though rather long (compare Three Dimensions,
pp. 559-567) affords some useful exercise in the manipulation of our formulae.
If q is the point of contact and qr the tangent touching at four consecutive
points, we nave
We may suppose the point r to lie in an arbitrary plane S^=0, and we
have to obtain the resultant of the four equations in r and finally to free it
from the arbitrary L Let Sra=0 and Sr6=0 be the equations of planes
through the generators of the quadric (r variable) Sr/r=0 which lie in the
tangent plane Srp=0. Thus we have r=[apl] and /=[fy?Z] for the points in
which three generators meet the arbitrary plane. One or other of these
points must lie on the cubic in r. Hence
SrD 3 .#.S/D' 3 .'=0, or SrD^.^.Sr'D 3 . =0,
or (SrD 3 . SKD^+SKD 8 . SrD' 3 ) '=0,
ART. isr.] . FOUR POINT CONTACT. 299
where the accents applied to D and Q are temporary marks connecting
operator and operand. Now this may be written in the form
s - SABC) '=0,
where ^SrDSr'D, 2flSrD'SrD+Sr'DSrD / , <7=SrD'Sr'D',
and it is easy to express the operators A, B and C in terms of the function/.
In virtue of the definition of the planes Sra=0, Sr6=0, we have identically
Srfr=SraSrb + SrpSrc,
where Src=0 is some plane. Hence we find on replacing r and r' in A, B
and C by [apt] and [bpt] that
B=S[plV]f[plV'l C=S[plV']f[pZD'].
Remembering that p=fq and that Sp=0, we have by Art. 146, Ex. 5,
Sql SqD'
Sql Slf~*l Slf" 1 !)' = S(DSql-lSqD)F(D'Sql-lSqD')j
SqD Slf~ l D SDf~ l T>'
with similar expressions for A and (7, where Fnf-^ is Hamilton's auxiliary
function. Writing for the moment e=DSql ISqD and remembering that
D and D' operate on Q and Q' solely and not on q as involved in the
structure of the operators, we proceed to expand and to operate on \
We have
B*Q' = (8eF& . Sql - Se^S^D') 3 . Q'
= S^D' 3 . Q f . SqP - 3m(m - l)(m - 2) . n(
because by the identities at the beginning of this example
since Se^=0 and S^D /3 . '=??i(m-l)(7n-2). =0.
We retain for a purpose the term in Seq.
In like manner
3 - S^D'(SD^D' . Q'SeFl
' . Q r ) SqP + SqI>'*(SeFI)' . Q'SIFI + 2SIFW .Q'.SeFl). Sql.
The term S^D' . SD'^D' . Q' may be reduced by writing for the moment
''', where as is easily seen 2Saa r =4. This term becomes
--l)(m-2). w, and hence we find
A BO. Q f =SeFe.SeFD'.SWFD'. Q'SqP
- m(m - l)(j - 2)n(4SeFeSeFlSqP
From these two relations we get, if e'=D'Sql-lSqD',
- 3.4 J BC r )Q'=(4S^V 3 - SBeF&eFe'BJFJ) . Q'
- 3m(m - l)(m - 2) . n . (SeFeSlFl -
and the last term vanishes because Seq=0. Now it will be observed that
the operator in the first term is precisely the same as the original operator
with D' substituted for D'S#Z - ZS0D'. This remark allows us to write down
the result of operating on QQ' in the form
3 - ZABC) '=(4SD^D /3 - 3SDFDSD T D'SD' J FD / ) . QQ' . Sql*
- 2m(m - l)(m - 2) . * . (SV'FV'SIFI -
300 PJ&OJECTTVE GEOMETRY. [CHAP. xvn.
the object of the retention of the term in $qe being now apparent. But
the term we have retained vanishes by the reduction we have already made
use of. Thus Sql 6 conies off as a factor, and the equation of the surface is
EXAMPLES TO CHAPTER XVII.
Ex. 1. A right line meets three fixed lines aa\ bb' and cc'. The locus
of the harmonic conjugate of the point of intersection on the third line with
respect to the points on the other two is the intersection of the planes
(bVcq)(aa'cc')+(aa cq)(bb'cc')=Q ; (bb'c'q)(aa'cc')+(aa'c'q)(bb'cc')=Q.
Ex. 2. The general equation of a quadiic through the conic
.
Find the value of I' in order that the quadric may be a cone having its
vertex at a and show that the equation of the cone may be written in the
Ex. 3. A plane aa'p is drawn through a fixed line aaf, and the lines in
which it meets the planes S^=0 and S^=0 are joined to the points b and
b respectively. The equations of the joining planes are
(qaa'p)Slb-(baa'p)Slq=0 and (qaa f p)Sl'b'-(b'aa'p)$l'q=0,
respectively, and when^ varies the locus of their intersection is the quadric
surface (q$lb-b$lq, qSl'V-b'Sl'q, a, a') = 0.
Ex. 1 The four faces of a tetrahedron pass each through a fixed point,
a, 0, c and d respectively. The three edges in the face p which contains the
point d he m the planes, I, m and n respectively. The vertex q opposite the
face p is the intersection of the planes
,
and the vertex q describes the cubic surface
(aSql-qSal, b$qn-qSbm, c&qn-qBon, d) = 0,
having the intersection of the fixed planes as a double point.
Ex. 5. Find the locus of the vertex of a tetrahedron, if the three edges
which pass through that vertex pass each through a fixed point, if the
opposite face also passes through a fixed point and the three remaining
vertices move in fixed planes.
rt. A Pi^^ 8868 ^ough a fixed point d, and the points in which
^ r ?f " ia * b ^ and C i c 2 are joined by planes to three
1168 ^ J "^ The 10CUS * in
iw^iirt, The f ? ide8 o a . P ol ygon pass through fixed points, a 2 , ...,
and all the vertices but one move in fixed planes, Z 1? Z,, ... l n _i If ^ the
" >^-^-^ft ind theUs of\ne free^rtex'
, q, o n ]=0.
Ex. 8. All the sides of a polygon but one pass through fixed points
extrltles V he f ^e side move on fixed fines 66' and at,
on fixed
ART. isr.] EXAMPLES. 301
Ex. 9. The points of contact q of tangent planes through the line ab to
the quadric Sqfq0 satisfy the relation*
fq=x[abq], where n+tf{$(Jafb)(ab)-S[fafb][ab]}=Q,
n being the fourth invariant of/, and if c is arbitrary
Ex. 10. If the line ab is a generator of the quadric
=a scalar.
(06)
Ex. 11. The generators of the family of quadrics S^(^/ 1 +y/ 2 -f3/ 3 )^=0
compose the complex of lines of the third order represented by "the deter-
minant equation
ISfcAfc $Pf n q, Sp/^|=0 (n = 1,2 or 3).
(a) When p is an arbitrarily selected fixed point, this equation represents
a cubic cone, and every edge of the cone determines a definite quadric of the
family.
,
ily. The tangent planes at p to the quadrics pass through the edge of
cone which joins p to the point [fip^ }%p, f B p] ; and the tangent plane to
the cone along this edge touches at the point p the quadric of which the
edge is a generator.
(b) When p lies on the Jacobian curve
the
the cubic cone breaks up into a plane and a quadric cone. The cone is a
member of the family of quadrics, and the plane touches at p all the
quadrics of the family which pass through p.
(c) The locus of points of contact of a plane Sfo=0 with quadrics of the
family is the cubic curve in which the plane cuts the surface
and the locus of points of contact of pairs of the quadrics is
[/ifc /sfc /*]-<>
Ex. 12. The integral of the differential equation
(<ty/S)-Oi or <ty=/.d*,
where /is a linear function, may be written in the form
q=e*.a,
where a is a quaternion constant of integration.
(a) This integral represents a doubly infinite family of curves, and a
determinate curve of the family passes through an arbitrary point provided
it is not a united point of the function/.
(b) The equation p = e~v . b
is the reciprocal of the tangent line developable of the curve determined by a
if the conditions
S6a=0, SW=0, S6/2a=0
are satisfied.
(c) An arbitrary plane which does not pass through a united point of /is
osculated by a single and determinate curve of the family.
*For another form see Art. 146, Ex. 5.
302 PEOJECTIVE GEOMETBY. [CHAP. xvn.
(d) An arbitrary tangent line to an arbitrary curve of the family is cut
in a constant anharmonic ratio at the point of contact and at the points of
intersection with three of the united planes of/.
(e) A. right line which cuts the faces of the tetrahedron in points having
a certain anharmonic ratio touches a definite curve of the family, and if
p and q are two points on the line
(Pitofafti'-O'
( f ) Any linear transformation which leaves unchanged the united points
of /j merely interchanges curves of the family.
(g) The locus of points of contact of tangent lines drawn from an
arbitrary point c to curves of the family is the twisted cubic
the locus of points of contact of tangent lines drawn through an arbitrary
line cd is the quadric (cdqfq)=Q ;
and the locus of points of osculation of planes through c is the cubic surface
fa frj
Ex. 13. The equation of the complex of lines cutting a tetrahedron in
points having a given anharmonic ratio may be written in the form
(P, ?, fP, /?)=<> where
h 4
is the given anharmonic ratio, t 19 2 , t$ and 4 being the latent roots of /and
the tetrahedron being determined by the united points of the function.
(a) The differential equation of curves whose tangents cut the tetrahedron
in points having the given anharmonic ratio is
(d?, ?,/d?,/2)=0;
and a solution of this equation is
f7?- d <
?=>' .a,
where a is an arbitrary quaternion and where u and v are functions of t*
(b) This equation includes the family of curves (compare Ex. 10, p. 286)
(c) In general the reciprocal of the tangent line developable of the
curve (a) is f *
~'
where S&a = S6/a = S&/% = 0.
(d) The anharmonic ratio of the point of contact and of the points in
which a tangent line to the curve (a) cuts the faces of the tetrahedron
corresponding to the roots t l9 t% and t 3 is
CHAPTER XVIII.
HYPERSPACE.
ART. 158. Many of the methods of quaternions are applicable
with but slight change to the general case of a " flat " space of
7i dimensions.
Commencing with the multiplication of two vectors or directed
lines in space of n dimensions, we may suppose the two vectors
to be transferred to one common plane or even to be made
coinitial, and we may define the product a/3 very nearly in the
same manner as in quaternions. In the formulae of definition
fa = - V 2 aj8 + Vft ............ (l.)
V a/3 or Sa/3 is minus the projection of one vector on the other
multiplied by the length of the latter, and V 2 a/3 is the directed
area of the parallelogram determined by a and /3, rotation in the
plane from a to /3 being positive. We can no longer identify
V 2 a/3 with a vector perpendicular to the plane because in space
of many dimensions there is an infinite number of directions
perpendicular to a plane.
In particular if i l3 i 2 ,-. in are n mutually rectangular unit-
vectors in the space of n dimensions, we have by (i.)
where s and t are any two numbers from 1 to n.
The functions V 2 a/3 and V p a/3 are doubly distributive, and
hence the binary product aft is doubly distributive. We define
for products of higher order that multiplication is thoroughly
associative and distributive, and these principles in conjunction
with (i.) form an adequate symbolical basis for the whole
calculus.
If \ and i 2 are any two mutually rectangular unit vectors in
the plane of a and /3, and if rotation from \ to i z is in the
same sense as that from a to /3, we may write
304 HYPERSPACE. [CHAP. xvm.
or in any parallel plane. This symbol ifa is of a distinct kind
from the symbols i v i 2 , ...i n , and it cannot be expressed as a
linear function of the latter.
In virtue of the laws of multiplication
and hence by (in.) the effect of multiplying a vector area into a
vector in its plane is to turn that vector through a right angle
in the plane and to multiply its length by the number of units
in the area.
For three vectors, which may be transferred to a common
space of three dimensions or even rendered coinitial, the laws of
the calculus allow us to write
, ..................... (IV.)
where V 3 a/3y denotes the part of the product depending on sets
of three distinct units combining in the irreducible products
ifaiv etc., and where Vxa/Sy arises from reducible products such
as i x s = i v \\ i& i{ifa=\. In fall if a = 2^, /3 Sj/^p
y=20 1 i 1 , where x, y and z are scalar coefficients, we find
}
. - Sa^Sz^, )
where | x^^ | denotes a determinant.
The first part V 3 /3y of the product of three vectors represents
the directed volume of the parallelepiped determined by the
vectors, it being now necessary to distinguish between volumes
in different spaces of three dimensions. In particular i-fai^
represents unit volume in the space of i I9 i 2 and i 3 . The
function V 3 a/3y is evidently combinatorial with respect to
the three vectors. It is unchanged when a is replaced by
a+vfi+wy, etc., and it changes sign when any two of the
vectors are transposed.
We have given the expansion for V^/Sy in terms of the unit
vectors and of the scalars x,y,z; but there is another method of
wide application which we may employ. It is apparent that we
must have
V ia /3y = UaV /3y + Vj8 V ay + Wy V a/3,
where u, v and w are numbers. Interchanging $ and y we have
V 1 ay/3 = UaV y / 8+t;y
adding and attending to (i.) we find
ART. 158.] GENERALIZATION OF FOBMULAE. 305
and thus i& = l, v+w*=Q. Similarly interchanging a and ft we
find that w = l and u+v = Q, and thus
j 8 ................ (VL)
By the same process we arrive at the result
aftyS^^aftyS + ^aftyS + ^aftyS ........... ( VIL )
for the product of any four vectors, where
V 2 /3y 8 = V 2 a/3V y S - V 2 ay V /3(5 + V 2 a5 V /3y + V OJ 8y \V<5
V a/3y S = V a/3 V y S - V ay V OJ 8<5 + V a<5 V /3y ;
and it will be noticed that in these relations the determinant
rule of signs is in every case obeyed, namely starting with^the
term ctV /3y, the next term, in which ft comes first, has a minus
sign and so on. In like manner for five vectors
) (lj x
) .........
the first terms in the sums being affected with the positive sign
and the determinant law of signs being obeyed. (Compare
Art. 147, p. 270.)
Considering more particularly the function of m vectors
V^^ ... a m , it is apparent from various points of view, that it
is combinatorial with respect to the m vectors. We may prove
that it changes sign whenever any two vectors are transposed,
and hence we may deduce the combinatorial property. Adding
the products
in the second of which a x and a 2 are transposed, ^ the sum ^ is
2a 3 a 4 . . . a m V io- * n this sum the higk est terms in the units
.are of the order m-2, and consequently interchange of con-
tiguous vectors changes the sign of V m aja 2 ... c^,... a m . Hence
transposition of any two vectors changes the sign ; for example
p I changes of sign accompany the transference of a p to the
first place in the function, andp-2 changes arise when a, is
transposed with a,, with a 3 and so on till it reaches the place
originally occupied by op. The function consequently vanishes
if any two vectors are identical, and when the vectors a are
replaced by vectors ft which are given as linear functions of the
a, the function is simply multiplied by the modulus of the
transformation.
J.Q.
306 HYPEESPACE. [CHAP. XVIIK
Generally any function such as
is combinatorial with respect to the vectors, and when we
express the m vectors a in terms of m mutually perpendicular
vector units in their m-space, we find that
ny\
= - Om = 2 V^tta . . a p V, n -^Op+iOp+2 - - - a- (XI.)
This includes a number of relations such as
3 V 3 a/3y = a V 2 /3y - /3 V 2 ay + y V 2 a/3 ;
Again when the m vectors lie in a space of m 1 dimensions
so that they are linearly connected, we have relations of the
- 1 V QfllM . / YTT v
" J ............... V A L1 ' r
which may be veiified by operating with V a 1} etc. In particular
for two and three dimensions, we recover the formulae, Art. 27
(in.) with S/>a/3 = and Art. 26 (n.).
The theory explained in this article may be compared with
Grassmann's Ausdehmmgslehre* Grassmann's inner product
of two quantities is the function V a/3, and his outer product
of op a 2 , . . . a m is V^a^^ . . . ^ These so-called products are thus
only parts of a complete associative product.
AKT. 159. There is a remarkable difference between this
general theory and the theory of quaternions which may be
illustrated by a special example. The sum of a number of
vector areas is not an area vector, or the homogeneous quadratic
function of the units
gyy'+etc ................... (l.)
cannot generally be expressed in the form V/>/o'. The geometrical
reason for this is that two planes, for example /> = ,T 1 'i 1 +a? 2 i 2 and
/> = 0^3+0^, have not necessarily a common line although they
may have a common point the origin of the vectors p in the
example.
To discover a canonical system of vector units in terms of
which a homogeneous function (q) of order m may be expressed,
observe that 9 f p=V m +i2/)+V m _ 1 g'p, and that the line vector
VjjVja-!^ is not generally parallel to p but that it is a linear
and distributive function of p. We are thus led to consider the
linear and distributive function
*See Proc. J.A. t 3rd Series, vol. vi., pp. 13-18 (1900).
ART. 159.] KOTATION" IN HYPERSPACE. 307
and because
V
.
= V . V m -i?o-V_ip2 = V .
the function is self -conjugate, and just as in quaternions its
axes are all real and mutually rectangular.
In particular for the quadratic A, let \ be an axis of
d>p = V^\Ap so that ^ = 6^, where a 12 is a scalar.
Then <j>V 1 Ai l = V l A^\A^\A^ = y l A<}>i l = a l z^ T l Ai l , and V^
is also an axis and it is perpendicular to i x and parallel to i E
suppose. This shows that in terms of the canonical units
A =
so that a quadratic in 2m 4-1 or in 2m units may be reduced to
m terms involving pairs of units, or to the sum of m area vectors.
There is obviously indeterminateness in the units to the extent
that i^ may be any unit in a definite plane that of ^ and i 2 ,
and i z may be any unit in another definite plane, and so on.
An expression such as A corresponds to an angular velocity
in the space of three dimensions. Consider the transformation
which converts line vectors (/>) into line vectors (<r = 0/>) and
which preserves unchanged lengths and mutual inclinations, so
If a is an axis of this function and * the corresponding root,
we have *V = V 0a 2 -V a 2 = a 2 ,
and therefore < = 1 or else <* 2 =0. The former alternative cor-
responds to non-rotate_d_directions. The latter requires a to
be of the form i 1 +V-l.^-a vector perpendicular to itselt
directed to one of the circular points at inimity in the plane
of *, and <L (Ex. 1, p. 96). Corresponding to this there is
a conjugate axis, a^-x/^l^ Again if ft is any other
axis corresponding to the root s,
so that axes, corresponding to roots which are not
one of the other, must be perpendicular From this
that the transformation is specially related to a **
perpendicular planes, ^ V* etc -' and ^V*
ordinary rotations in each of these planes, so that
we may
where 2 12S4 -,
and where the factors q^ q w etc, are commutative because we
308 HYPEBSPACE, [CHAP. xvm.
Also we have
It may at once be verified that the operator g 12 ( )g 12 "" 1 has no
effect except on vectors in the plane of i^g, and that it turns
vectors in this plane through the angle a 12 .
Now we may write (Art. 29 (v.), p. 28)
fia
and because the factors are all commutative we may also write
i(ai2^2+a84^4+..) $A
q = e = ..................... (v.)
(compare (ill.) and Art. 29 (x.)), and the rotation is effected by
jji \A
the operator e ( )e
For a small rotation, if dt is a small scalar whose square is to
be neglected,
and thus A plays the part of an angular velocity.*
ART. 160. For projective geometry in n space we may use
the method explained in the last chapter, and the symbol for a
point is the sum of a scalar and a vector, so that
.......... (I.)
represents a point of weight V g at the extremity of the vector
e equation g^+^+efa, ... + t m a m ................... (n.)
represents the (m-l)-flat which contains the points a 19 a 2 , ... a m .
In accordance with Hamilton's notation, we shall write"
. . . a m ] = V w . V^V^ . . . V 1 a fB
or briefly, M w = V w [a] m +V i;l . 1 [a] m , .................. (m.)
as the symbol of the (m l)-flat containing the m points a. To
show that this symbol really determines the flat, observe that
we have
where a^a^V^ and where II V^ is the product of the
weights of the points (Art. 144 (iv.)). Now V^a 1 a 2 ...a m or
V^a^aj a x ) ... (a w a x ) is the directed region determined by the
origin and the m points, and V w _ 1 (a 2 -a 1 )(a 3 -a 1 ) ... (a m -a x ) is
*See Proc. R.I.A., Series HL, vol. ^ pp. 7^-123.
ART. loo.] PKOJECTIVE GEOMETRY. 309
that determined by the m points. Denoting the latter by
iji^ ...i m -iR, where R is the magnitude of the region determined
by the m points and where ^ 1 i 2 .-.fy~i are mutually perpen-
dicular unit vectors in the flat, the symbol becomes
(v.)
where or is the component of the vector 04 which is perpendicular
to every line in the flat, or in other words, where trr is the vector
perpendicular from the origin to the flat. But when we know
trr and the product of the -vectors i we know the flat,* and we
have r-,
V m - 1 L a Jm
where U has its quaternion signification. We notice also that
the point v , fV/1
P 1 I m " lL Jm 1 rr-l fVTl^
+ ~T~Tffll ............... ^ '
mL^Jm
is the reciprocal with respect to the auxiliary quadrie V g 2 =0 of
every point in the flat in other words, this point is the point
in the m-flat of the origin and of the m points a which is
reciprocal to the (m-l)-fiat of the points a.
In point symbols the equation of the flat is
the vanishing of this equation being equivalent to (n.).
Other general expressions admit easily of interpretation on
the principles laid down in this article.
*The vector equation of the flat is p=5T-f Sar^.
INDEX.
The Numbers refer to pacjes. When a chapter is quoted, the Table of
Contents may be consulted with advantage.
Aberration, 85.
Academy, Royal Irish, 101, 152, 163,
164, 263, 306, 308.
Acceleration, trajectory of point under
uniform, 64; relative, 171 ; angular,
171; centre, 172; of a particle, 184
et seq. ; of a rigid body, 194 et seq.
Activity of forces on element of strained
medium, 240 ; of electric and mag-
netic forces, 251.
Addition of vectors, 1 ; of quaternions,
9 ; of vector-arcs, 16 ; of weighted
points, 264 ; of vector areas in hyper-
space, 306.
Algebra, vectorial, 11.
Algebraic sign - , use of, 2.
Algorithm of i, j, k, 11 ; for hyper-
space, 303.
Almucantar, example on, 176.
Amplitude of versor, 27.
Analytical expressions for V, 74, 225.
Anchor-ring, 59.
Angle of quaternion, 13; differential of,
69 ; directed, 30 ; Eulerian, 33 ; of
intersection of spheres, 50 ; element
of solid, 86 ; of contact, 134 ; of tor-
sion, 134 ; subtended by vortex ring,
solid, 235.
Angular acceleration, 171 ; momentum,
184, 195; velocity, 170, 187; of
emanant, 132; of strained element,
212 ; in hyperspace, 307.
Anharmonic coordinates, 43 ; equation
of sphere in, 54; in relation to
weighted points, 269 ; ratio of collin-
ear points, 41, 45 ; of four points in
space, 56 ; of points on a conic, 267 ;
generation of hyperboloid, 65; pro-
perties of ruled surface, 139 ; of
twisted cubic, 268 ; unaltered by
linear transformation, 272 ; complex
of lines cutting faces of tetrahedron
in constant, ratio, 302.
Anisotropic medium, 243, 251.
Apparent double points, 292.
Appendix to uew edition of Memento of
Quaternions, 99, 135, 211.
Arc, vector-, 17; cyclic, 118; of curve,
134.
Area, directed, 23 ; of spherical triangle,
33 ; -vector in hyperspace, 303.
Areal coordinates, 48 ; velocity, 186 ;
equation of continuity, 230.
Aronhold's notation, 213, 297.
Aspect of plane, 23.
Associative addition of vectors, 2;
multiplication of i t j, k, 10 ; of quat-
ernions, 11, 119 ; of vectors in hyper-
space, 303.
Astatics, 160.
Astronomy, examples from, 84, 85, 130,
174, 188. "
Asymptote of conic, 64 ; of curve, 152.
Asymptotic cone, 107 ; lines on surface,
Attraction to fixed centre, particle
moving under, 185, 186; Green's
theorem, 218 ; spherical harmonics,
222.
Ausdehnunyslehre, 306.
Auxiliary functions, x and ^, 90, 91 ;
F, G and H, 274 ; quadric, 266 ; for
hyperspace, 309.
Axes of linear vector function, 94 ; of
self -con jugate function, 96 ; of quad-
ric, 111 ; of section of, 111 ; of screw-
systems, 163; moving, 171 ; for curve,
134; for surface, 146; for orbit, 188;
for body, 196 ; of inertia, 197 ; of
elastic symmetry, 245.
Axis of quaternion, 13 ; radical, 51 ;
Poinsot's central, 156, 169; instan-
taneous, 170.
Ball, Sir R. S. s theory of screws, 156,
163, 170, 203, 205.
INDEX.
311
Base-point, 171, 195.
Bilinear function, 297.
Binet's theorem on axis of inertia, 197.
Binomial, 134.
Biquadratic equation of linear quater-
nion function, 274.
Biquaternions, 20, 58.
Bisecting sides of spherical triangle,
triangle, 31.
Body, rigid, dynamics of, 196 et seq. ;
under no applied forces, 198 ; dyna-
mically equivalent to four particles,
199; dynamical constants of, 199,
202, 207; impact of, 203; con-
strained, 204; resultant force and
couple on gravitating, 225 ; moving
in fluid, 241.
Bonnet's theorem, 192.
Brachistochrone, 192.
Bright curves on surface, 87.
Bulkiness of fluid, 240,
Burnside, theory of groups, 104.
Calculus, icosian, 104; of variations,
192.
Canonical, form of V, 75 ; of two linear
functions, 100 ; of screw-systems,
164; equations of quadric and linear
complex, 278 ; vectors for rotation in
hyperspace, 307.
Cavity filled with liquid, motion of body
containing, 241.
Central, sections of quadric, 111 ; sur-
faces, non-, 117; axis of forces, 156,
163; of displacement, 169; orbit, 186.
Centre, mean, of tetrahedron, 5 ; of
mass, 5, 264; of circle inscribed to
triangle, 48 ; radical, 52 ; of quadric,
117; of curvature of curve, 134; of
spherical curvature, 136; locus of
mean, of corresponding points, 152;
of forces, Hamilton's, 157 ; astatic,
160 ; of three-system of screws, 164 ;
particle attracted to, 185, 186. .
Centres, of curvature of quadric, 122;
surface of, 125 ; of surface, 144 ; of
generalized curvature, 286, 295.
Chain on surface, equilibrium of,
166.
Characteristic surfaces in optics, 228.
Characteristics of curves and surfaces,
numerical, 290.
Charpit's differential equations, 151.
Chiastic homography, 208.
Circle, inverse of line, 53 ; at infinity,
imaginary, 54; monomial equation
of, 55 ; quaternion equation of, 58 ;
vector equation of, 82 ; ellipse pro-
jected into, 83 ; osculating, 134, 136,
152; surface generated by, 154;
excluding point from integration,
217.
Circuit, integration round, 73, 215 ;
circulation and flux, 232 ; moving in
perfect fluid, 238 ; electro-magnetic,
249.
Circuitation equations for electro-mag-
netic field, 250.
Circular, points at infinity, 96, 126 ;
sections of quadric, 113 ; of cone,
118; in relation to strain, 178;
tangent cylinder. 115 ; point at in-
finity in hyperspace, 307.
Circulation of vector, 232.
Circumscribed developable of confo-
cals, 126 ; generalized, 286.
Clifford, biquaternions, 21.
Coaxial, spheres, 51, 53 ; linear vector
functions, 95, 97 ; stress and strain
functions, 238.
Co-efficient, differential, 63, 67 ; of
friction, 190; of restitution, 204;
virtual, of screws, 206 ; of viscosity,
239.
Coelostat, example on, 130.
Coincidence, of axes of function, 94 ; of
united points, 275.
Collinearity, of three points, 5, 37,
266 ; of three planes, 39.
Collision of two bodies, 203.
Combinatorial functions, 265, 270, 304.
Commutative, addition of vectors, 1 ;
multiplication, 17 ; order of differ-
entiation, 79 ; linear functions, 95 ;
small displacements, 169 ; strains, 182.
Complementary curve, 291.
Complex, or imaginary, 3, 20, 58 ; th
roots of quaternion, 28 ; of right
lines, 40 ; surfaces formed by lines of,
153 ; related to astatics, 161 ; of axes
of inertia, 197 ; linear, 275 et seq. ; of
lines connecting corresponding points,
278 ; of generators of systems of
quadrics, 301 ; tetrahedral, 302.
Composition, of wrenches, 164, 204;
of displacements, 168.
Concurrence of four planes, 39, 267.
Con cyclic quadrics, 121.
Conductivity, electrical, 251.
Cone, tangent to sphere, 49 ; to quadric
108 ; to confocal, 124 ; standing on
curve, 65 ; of axes of system, fa + tfa,
101 ; asymptotic, 107 ; edges of, in
plane, 110; and sphero-conic, 118;
through five lines, 121 ; of revolu-
tion through three lines, 126 ; differ-
ential equation of, 149; tangent to
generalized confocal, 281.
Confocal, quadrics, 121 ; tangent cones,
124 ; vector equation of, 124 ; re-
lated to astatics, 162; related to
axes of inertia, 197 ; oquipotential
system, 228; generalized confocald,
279 ; quaternion equation of, 286*
312
INDEX.
Congruency of lines, 41 ; surfaces
generated by lines of, 153; focal
and extreme points, 153 ; of axes
of three-system of screws, 164,
103.
Conic, related to triangle, 48; vector
equations of, 63; focal, 114, 126;
sphere-, 118 ; and Pascal hexagon,
121; orbit, 187; on wave-surface,
261 ; in point symbols, 264, 267 ;
anharmonic property of, 267.
Conical refraction, elastic solid, 248;
dielectric, 260.
Conical rotation, represented by
q ( }q" 1 i 18 ; related to spherical
triangle, 32; in terms of Euler's
angles, 33 ; inscribed polygon, 55 ;
examples, 60; differential of qaq~ l 3
69, 169 ; and linear function, 100 ;
and astatics, 160; finite displace-
ments, 168 ; examples, 173 ; strain,
178 ; and linear quaternion function,
283 ; in hyperspace, 307.
Conicoid, see Quadric.
Conjugate, of quaternion, 12 ; of pro-
duct, 15 ; radii of conic, 63 ; of linear
function, 89 ; axes of function and of
its, 94 ; quadric, 107 ; radii of qua-
dric, 110, 112; of quaternion function,
273, 275 ; tangents, 295.
See Self -conjugate.
Connected region, 217.
Conservative system of forces, 184 ;
acting on perfect fluid, 238.
Constant, curve having ratio of curva-
ture to torsion, 137.
Constants of linear function, 88, 178 ;
vector, of integration, 137, 186 ; dy-
namical, of rigid body, 199, 202, 207;
elastic, 239, 244 ; dielectric, 251 ; of
quaternion function, 272.
Constrained motion of particle, 189;
of rigid body, 204.
Construction of product of two quater-
nions, 15 ; fourth proportional to
three vectors, 31 ; ellipsoid, 114 ;
vectors related to wave in dielectric,
259.
Contact, of line and sphere, 49 ; and
quadric, 107 ; and confocals, 124 ;
four point, of tangent, 298.
Continuity, equation of, 72, 230, 238 ;
areal and linear, 230.
Convention respecting rotation, 7 ; nota-
tion, 20.
Convergence of vector, 72, 212.
Co-ordinates, six, of a line, 40 ; anhar-
monic, 43, 48, 54, 269; curvilinear,
66, 74, 227 ; Cartesian, 75 ; elliptic,
124, 286; homogeneous or tetrahedral,
268*
Coplanar versors, 27.
Coplanarity of four points, 5, 38 ; in
point symbols, 266.
Co-reciprocal screws, 206.
Co-residuals on cubic, 101.
Correspondence, ,see Homographic,
Transformation.
Co variant linear functions, 101, 290.
Cremona transformation, 101.
Cross ratio, see anhannonic.
Crystalline medium, damped oscilla-
tions in, 186 ; propagation of light
in, 256.
Cubic, of linear vector function, 93, 100;
twisted, 93, 104; cone, 101; twisted,
locus of feet of normals, 109 ; of
points of contact with confocals, 123 ;
tangent line and osculating plane,
133; related to moving body, 172;
developable generated by, 267 ; locus
of points in perspective with corre-
spondents, 278 ; transformation of,
285 ; characteristics of, 293.
Curl of vector, Wo-, 73, 213.
Current, electric and magnetic, 250. -
Curvature, of curve, 132 et seq. ; of
surfaces, 141 et seq., 215 ; of quadric,
122, 125 ; of orbit, 189 ; generalized,
286, 295.
Curve, in terms of parameter, 62 ; of
intersection of confocals, 125 ; me-
trical properties of, 131 et seq, ; uni-
cursal, 152 ; intersection of quadrics,
285 ; complementary, 291 ; character-
istics of, 292.
Curves, family of, 148 ; q = (/+ t) m a t 286;
q = <P.a, 301.
Curvilinear coordinates, 66, 74, 124,
226.
Cusp, condition for, 63, 83.
Cuspidal edge, 126, 136, 268, 286.
Cyclic planes of quadric, 113, 178; arcs
of sphero-conic, 118.
Cyclical transposition under sign S, 16.
Cycloid, 83, 193.
Cylinder, right circular, 45 ; standing
on curve, 65 ; circular tangent, to
quadric, 115 ; case of general quadric,
117 ; geodesic on, 137; torsal tangent
planes of, 140 ; differential equation
of, 149 ; related to astatics, 161.
Cylindroid, 84, 165.
D symbol of differentiation, 229; of
operator analogous to V, 296.
Damped oscillations, 186.
Deformation of surfaces, 145,
Degraded, cases of quaternions, 9, 19 ;
symbolic equations, 95, 275.
Degree, see Order.
Degrees of freedom, 204.
Delta, Hamilton's operator V, 70, 211.
See Operator.
INDEX.
313
De Moivre's theorem, 27.
Derivative, 63.
Determinants and combinatorial func-
tions, 270, 305.
Developable surface, 65 ; circumscribing
confocals, 126 ; related to curve, 135,
139; generated by tangent planes
along curve on surface, 142; of twisted
cubic, 267 ; circumscribing quadrics,
280; tangent-line, of two quadrics,
282; circumscribing generalized con-
focals, 286.
Development of quaternion function,
79, 85 ; of vector of curve in terms
of arc, 134.
Deviation from osculating curve, 152.
Diaphragm, 217.
Dielectric, 251 et seq.
Difference of two points, 263.
Differential, 63, 66 ; condition for per-
fect, 74, 86, 214 ; indeterminate, 87 ;
of equation of surface, 142 ; equation
of geodesic, 141, 152; of lines of
curvature, 144, 147 ; of family of
surfaces, 149 ; of curves traced on
surfaces, 287.
Differentiation, chap. vn. , 62 ; general
formula, 66 ; successive, 79 ; with
respect to moving axes, 167 et seq. ;
of deformable elements, 212 ; follow-
ing moving point, 229.
Diffusion of electromagnetic disturb-
ance, 255.
Dilatation in strain, 178.
Direct and inverse similitude, 14.
Directed area, 23 ; angle, 31 ; curva-
ture, 132, 141 ; volume in hyperspace,
304.
Discontinuity in integration, 216.
Displacement, of a body, 18, chap, xn.,
168 ; in strain, 180 ; electric, 250.
Dissipation function, 240, 252.
Dissociative multiplication, 11.
Distortion of elements, 212, 229; of
viscous fluid, 238.
Distributive, multiplication of vector
by scalar, 4 ; by vector, 8 ; property
of scalar of product, 6 ; of product,
9 ; of differential, 66 ; of linear func-
tion, 88 ; multiplication for hyper-
space, 303.
Disturbance in electromagnetic field
". by waves or by diffusion,
Divergence of vector, 212.
Division, of vectors reduced to multi-
plication, 11 ; homographic, 41, 65,
152, 264.
Dodecahedron, 104.
Double points, on wave surface, 248,
261 ; apparent, 292; on Jacobian, 298.
Duality for point symbol, 265.
Dynamical constants of a body, 199,
202,207.
Dynamics, of a particle, chap, xiv.,
184 ; of system and rigid body, chap.
xv., 194; of continuous medium, 236;
electro-, 249.
Eight square roots of linear function,
99 ; umbilical generators, 125 ; gen-
eralization of, 279, 286; generators
which are also normals, 279.
Elastic solid, iso tropic, 222, 239; aniso-
tropic, 242 et seq. ; symmetry, 245.
Electro-magnetic theory, 249 et seq. ;
of light, 256.
Element, rate of change of, 212, 229.
Elements of Quaternions referred to,
1, 3, 7, 29, 31, 34, 45, 53, 55, 56, 59,
82, 85, 114, 118, 120, 121, 132, 156,
157, 197, 211, 264; appendix to, 99,
135, 211,
Elimination of a vector, 39, 105.
Ellipse, vector equation of, 63, 82 ; pro-
jected into circle, 83; parallactic, 85;
aberrational, 85 ; differential equation
of surface generated by, 149 ; related
to astatics, 163 ; locus of feet of per-
pendiculars on generators of cylin-
droid, 166 ; in conical refraction, 261.
Ellipsoid,* Hamilton's construction for,
114 ; vector equation of, 152 ; strain,
177.
Ellipsoidal linear function, 178.
Elliptic, logarithmic spiral, 82 ; co-
ordinates, 124 ; functions, 198 ; gen-
eralized, co-ordinates, 286.
Elongation, 181.
Emanant, 131, 138.
Energy equation, for particle, 184, 187;
system of particles, 194 ; rigid body,
197 ; for impulses, 200 ; for contin-
uous medium, 239; in electro-mag-
netic theory, 251 ; function, for elastic
solid, 243 ; for dielectric, 252.
Envelope, examples, 128, 129 ; differen-
tial equation of, 149, 151 ; wave-
surface as, 248, 257.
Epicycloid, 83.
Equality of vectors, 1 ; vector-arcs, 17 ;
points, 263.
Equilibrium, static, 156 ; astatic, 160.
Equipotential surfaces, 227.
Euler's angles, 33 ; four square identity,
16 ; exponential formulae, 28 ;
theorem on curvature, 143 ; equations
of motion of rigid body, 196 ; of fluid,
230, 238.
Evoked wrench, 201.
E volutes on polar developable, 139.
* See Linear vector function, the use of an
ellipsoid being to a great extent superseded.
314
INDEX.
Exact differential, 74, 86, 214.
Excentricity of orbit, 187.
Excess, spherical, 33.
Expansion, of quaternion function, 79,
So ; of vector of curve in terms of arc,
134 ; in series of spherical harmonics,
223, 224.
Exponential pf quaternion, 28, 34;
differential of, 86; for hyperspace,
308.
Extreme points on line of congruency,
154.
Families of curves and surfaces, 148.
Family of equipotential surfaces, 227 ;
of curves, q=(f+t) m a, 286; q=e*Sa,
301.
Five vectors, 43, 44, 54; quaternions,
43, 269 ; points linearly transformed
into five, 272 ; surfaces, 291.
Flat space, 303 ; symbol of, 308.
Flow of a vector, 231.
Fluid, motion, 72, 229, 236; viscous,
238, 240 ; motion of solid in, 241.
Flux through circuit, 233 ; strength of
tube of, 233, 235 ; in electro-magnetic
theory, 249; of radiated energy,
Poynting, 252, 257.
Focal, property of quadrics, Salmon's,
114; form of equation, 116; for
sphero-conic, 120 ; conies on develop-
able, 126 ; points on line of con-
gruency, 153 ; conies related to
astatics, 162.
Foci of central sections of quadric, 129.
Force, moment of, 23 ; in statics, 156 ;
in dynamics, 184, 194 ; central, 186 ;
impulsive, 200 ; electric and magnetic,
251; in electro-magnetic field, me-
chanical, 255.
Forces, reduction to two, 158 ; con-
servative, 184, 238 ; of interaction,
, 194 ; system of forces, see Wrench.
Formula, A, 11 ; B, 8 ; of differentiation,
66.
Formulae, depending on products of
vectors, chap, in., 23; of trigono-
metry, 25, 30.
Four numbers involved in quaternion,
9 ; squares, identity connecting, 16 ;
vectors, identities connecting, 24 ;
symmetrical relations for, 42 ; linear
function rendering four vectors par-
allel to, 92 ; particles equivalent to
rigid body, 199 ; -system of screws,
206 ; consecutive points on tangents
to surface, surface through, 298.
Fourth proportional to- three vectors,
31.
Fractions, relations reduced by partial,
Freedom, degrees of, 204.
Fresnel, 163, 262.
Frictional constraint, 190.
Function, anharmonic, of collinear
points, 41, 45, of points in space, 56,
on a conic, 267 ; linear vector, 88 ;
elliptic, 198 ; dissipation, 240 ; energy,
for elastic solid, 243, for dielectric,
252 ; combinatorial, 270, 304 ; linear
quaternion, 272.
See Linear function.
Fundamental formulae of trigonometry,
plane, 25 ; spherical, 30.
Gate, self-closing, 207.
Gauss, operator, 104 ; measure of cur-
vature, 144, 147 ; integration theorem,
215.
Generalised, normal, 279; curvature,
286, 295 ; geodesic, 287.
Generation of ruled quadric, 65 ; of
ellipsoid, 114 ; of ruled surface, 137.
Generators of quadric, 103, 116 ; um-
bilical, 125 ; common, and of linear
complex, 278 ; generalized umbilical,
279, 286 ; eight, are also normals,
279 ; complex of, of doubly infinite
family of quadrics, 301.
Geodesic on cylinder, 137 ; differential
equation of, 141, 152; curvature, 141,
148; Joachimstal's theorem, 152;
motion of particle along, 190 ; gen-
eralized, 287.
Geometrical meaning of invariants, 98,
288.
Geometry of Three Dimensions, Salmon's,
291, 292, 298.
Geometry, protective, chap, xvn., 263,
308.
Gilbert's theorem on confocals, 124.
Grassinann, 306.
Graves, R P., Life of Hamilton re-
ferred to, 16,211.
Gravitating body in field of force, 225.
Green's theorem adapted to quaternions,
219.
Groups, theory of, examples relating to,
80 ; referred to, 104.
Half-line, half-cone, 45.
Harmonic, mean of two vectors, 41, 50,
56, 109 ; properties of triangle, 45,
of polar and quadric, 50, 109 ;
spherical, 70, 76, 222 et seq.
Hathaway, A. S., 270.
Heaviside, Oliver, 11, 249, 250, 253.
Helix, vector equation of, 64, 82 ;
vector twist of, 133 ; constant curva-
ture and torsion, 137 ; osculating,
152 ; particle moving on, 191.
Herpolhode, 198.
Hessian of surface, 297.
Hexagon, Pascal, 121.
INDEX.
315
Higher Plane Curves, Salmon's, referred
to, 101, 105.
Hodograph, 83, 187, 189.
Homographic, ranges, 41, 42, 264 ;
locus of line joining corresponding
points, 65 j locus of mean centre of
points on, 152 ; screw-systems, 208 ;
correspondence of points on twisted
cubics, 285.
Homography, chiastic, 208.
Hooke's law, 243.
Hydrodynamics, 72, 228 et seq.
Hyperbola, 64; section of quadric,
rectangular, 111 ; focal, 114.
Hyperboloid, homographic generation,
65, 264 ; locus of transversals, 103,
270; generators of, 116; line of
striction of, 140 ; equilibrating forces
on generators of, 158.
Hyperspace, chap, xvni., 303.
Hypocycloid, 83.
Hysteresis, 251.
Icosian calculus, 104.
Identity, Euler's four square, 16 ; con-
necting four vectors, 24 ; five quater-
nions, 269.
Ikosahedron, 104.
Imaginary, of algebra, 3, 20, 58 ; ?i th
roots of quaternions, 28 ; roots and
axes of linear function, 95, 96, 177 ;
conjugate, vectors, 95, 224, 307 ;
united points of linear transforma-
tion, 275, 276.
Impact of two bodies, 203.
Impulse, 200.
Impulsive wrench, 201, 204 ; genera-
ting motion of solid in fluid, 241.
Indeterminateiipss of versor of null
quaternion, 19 ; of tensor of bi-
quaternion, 21 ; of a differential, 87 ;
in solution of equations, 92 ; of axes
of linear function, 95, 96 ; of square
roots of function, 99 ; in value of
function, 216 ; related to conical
refraction, 248, 260; of normal to
Slane in hyperspace, 303.
ex-surface, 248, 261.
Induction, magnetic, 250.
Inertia function for rigid body, 196;
Binet's theorem on axes of, 197;
deduced from observed motion, 199,
202, 207 ; related to Jtfdw?, 225.
Infinites in field of integration, 216, 219.
Infinity, anharmonic equation of plane
at, 44, of circle at, 54 ; vector to
circular points at, 96, 126, #07 ;
vector representing point at, 263 ;
equation of plane at, 266.
Inflexion on curve, 83.
Initial positions in astatics, 160.
Inscription of polygon to sphere, 55, 56.
Instantaneous twist-velocity, 170, 201 ;
orbit, 188.
Integrability, condition of, 74, 86, 214.
Integrals, line, 73, 215, 219, 231 ; sur-
face, 72, 215, 219, 233 ; variation of,
192, 231, 233.
Intensity of wrench, 163.
Interaction of particles, 194, 200, 236.
Interpretations and formulae, chap in.,
23; for projective geometry, 263
et seq.
Intersection of, line and plane, 35, 267,
269 ; planes, 39, 267, 269, 306 ; two
lines, 39, 267 ; line and sphere, 49 ;
spheres, 50, 54 ; confocals, 121, 123,
125 ; quadrics, 285 ; generalized con-
focals, 286 ; curve and complemen-
tary, 292 ; of two surfaces, osculating
plane to curve of, 296.
Invariants, of linear vector functions,
91, 97 ; geometrical meaning of, 98 ;
of two functions, 100 ; derived by
operation of V, 102 ; depending on V,
211 ; of linear quaternion function
274; of quadrics and linear trans-
formations, 288.
Inverse, or reciprocal of vector, 1 1 ; of
product 12 ; similitude, 14 ; trans-
formation, 90 ; operations of V, 218.
Inversion, geometrical, 52, correspond-
ing elements in, 69 ; of linear func-
tions, 90 ; of <j> + t<t>, 100 i of V, 218 ;
of linear quaternion function, 273.
Involution on ruled surface, 140.
Irrotational distribution of vectors,
234.
Isothermal surfaces, 227.
Isotropic solid, 222, 239.
Jacobj, differential equations, 86.^
Jacobian, or functional determinant,
213 ; of four quadrics, 293 ; of sur-
faces, 295, 298.
Joachimstal's theorem on geodesies,
152.
Joulian waste of energy in electro-mag-
netic field, 252.
K, symbol for conjugate, 12 ; differen-
tial of K#, 68.
Kelvin, Lord, flow along curve. 231.
Kinematical treatment of curves, 134;
of surfaces, 137, 145.
Kinematics, chap, xn., 168; of con-
tinuous medium, 228.
See also Motion.
Kinetic energy, of particle, 184, 187 ;
of system of particles, 394; of rigid
body, 197 ; changed by impulse, 201,
207; of portion of continuous med-
ium, 239.
316
INDEX.
Kinetics of a particle, 184; of rigid
body, 194; of continuous medium,
236.
Knott, C. G., 11.
Lagrange, motion of fluid, 230, 238.
Laplace's operator, 75, 227; inversion
of, 220.
Latent roots of linear function, 93, 96 ;
linear quaternion function, 274, 276.
Lectures on Quaternions referred to, 1,
7,21,59, 114,115, 121,211.
Lie, Sophus, 86.
Light, electro-magnetic theory, 256.
See Optics.
Limiting, points of coaxial spheres, 51 ;
ratios, 63.
Line, chap, v., 35; six coordinates of
40 ; inverse of, 53 ; of striction, 138,
140 ; of curvature, 144 ; in point
symbols, 266 ; unaltered by linear
transformation, 272; traced on sur-
face, 287, 295.
See Complex, Curve, Generator, In-
tegral.
Linear, relation connecting four vectors,
5, 24, 25 ; and distributive function,
66; vector function, chap. VIIL, 88;
related to quadrics, chap, ix., 106 ;
to surfaces, 142 ; to astatics, 159 ; to
theory of screws, 164, 205 ; to accel
eration of point of body, 172; to
strain, 177 ; to vibrations of particle,
186 ; to angular momentum of rigid
body, 196 ; to operator V, 211 ; to
stress, 237, 243 ; to electro-magnetic
field, 251 ; to theory of light, 258 ;
equation of continuity, 230 ; relations
connecting five quaternions, 268 ;
quaternion function, 272 et seq. ;
complex, 275; transformation, in-
variants of, 288.
Logarithm of a quaternion, 29.
Logarithmic spiral, 82,
Lorentz, H. A., 229, 249, 251.
Lunar theory, example on, 188.
M'Aulay, A., 21, 211, 218.
MacCullagh, index-surface, 248.
Magnetic force, 249; permeability, 251.
Maximum and minimum, 80, 111, 127.
Maxwell, J. Clerk, sense of rotation, 7 ;
curl of vector, 213 ; electro-magnetic
theory, 249.
Mean, point, 5; harmonic, of two
vectors, 41, 50, 56, 109; centre of
corresponding points, 152 ; in point
symbols, 264.
Measure of curvature, 144, 147.
Mechanical force in electro-magnetic
field, 251.
Medium, continuous, 228, 236, 251.
Meusnier's theorem, 141.
Minchin, 183.
Minding's theorem, 162.
Moivre's, de, theorem, 27.
Moment, of force, 23 ; resultant, 156 ;
quaternion, 157, 159 ; of momentum,
184, 195, 196 ; of inertia, 196.
Momentum, 184; moment of, 195, 196;
of portion of medium, 236 ; of solid
and fluid, 241.
Monomial equations of circle and sphere,
55.
Motion, three-bar, 60, 85 ; of point on
curve, 62; generating roulette, 83, 84;
apparent, 84 ; relative, 171, 174; of
body under no forces, 198 ; of con-
tinuous medium, 228, 236.
Moving axes, 171; for curve, 134; for
surface, 146; for orbit, 188; for body ,
196 ; for electro-magnetic field, 253.
Multiple- valued function, 216 ; point
on Jacobian, 298.
Multiplication, by scalars, 3 ; distribu-
tive, 9 ; associative, 11 ; of versors,
versor-arcs, 16 ; symbolical, table for
S,V,K,T,U, 19; hyperspace, 303 ; in
A wdeh nun(jsUhre> 306.
Mutual potential, 223.
Mutually rectangular vectors, system
of three, 10 ; relations connecting two
systems, 33; axes of function, 96, 97 j
vectors transformed from, 98 ; nor-
mal to confocals, 123 ; related to curve,
134; to surface, 146; examples relat-
ing to, 173.
Negative unity, square of unit vector
is, 10, 17; square-root of, 3, 20, 58 ;
see Imaginary.
Non-central surfaces, 117.
Non-commutative, multiplication, 8 ;
addition, 16 ; displacements, 168.
Nonion, see Linear vector function.
Normal, to surface, 65, 139, 144 ; to
quadric, 108, 123; to curve, 134;
and tangential resolution of force,
185, 189 ; solutions, 256 ; generalized,
279 ; generator as well as, 279.
Notation, conventions respecting, 19 ;
for projective properties of surfaces,
296.
See Symbol.
Nullifier, 21.
Number of constants of linear func-
tion, 88, 178, 272, 283.
Numerical characteristics, order of
cone and surface, 101 ; of curves, 290.
O'Brien, Rev. M., 11.
Octahedron, regular, 45, 104.
Octonions, 21.
Ohm's law, 251.
INDEX,
317
Operator, quaternion as, 14 ; V, 70 et
seq., chap, xvi., 211 ; various ex-
pressions for V, 74, 225 ; applica-
tions of V to Taylor's theorem, 79 ;
to theory of groups, 80, 86 ; to genera-
tion of invariants, 102; to strain,
181 ; to calculus of variations, 192 ;
to curvature, 215 ; V 2 , 75, 220, 227 ;
V" 1 and V- 2 , 218; Gaussian, 104;
D, analogue of V for protective
geometry, 296.
Opposite of vector, 1.
Optics, examples from, reflection, 19 ;
refraction, 22 ; aberration, 85 ;
astronomical refraction, 85, 175;
bright curves, 87 ; rotating mirror,
130 ; characteristic surfaces in, 228 ;
electro-magnetic theory of, 256.
Orbit, 186 ; instantaneous, 188.
Order of surface, 101 ; of curve, 290 ;
of multiple points on Jacobian, 298.
Origin, variable, 157.
Orthogonal spheres, 51, 52, 54 ; con-
focals, 121, 123, 125 ; surfaces, 227.
Oscillation of particle, 185; of rigid
body, 207.
Osculating plane, 132, 267, 296 ; circle,
134, 136, 152; sphere, 136; quad-
ric, 144 ; helix, 152 ; curve of inter-
section of two surfaces, 296.
Parabola, 64, 267.
Paraboloid, condition that general
equation should represent, 117;
related to constrained motion, 191.
Parallax, 85.
Parallelepiped, volume of, 22 ; integra-
tion over faces of, 71.
Parameter, vector involving, 62, 64,
65 ; form of V suitable for, 74, 226 ;
parameter of distribution, 138.
Partial differentiation, 67, 229; frac-
tions involving linear functions, 122 ;
differential equations, 86, 148, 151,
153 ; involving V, 226.
Particle, dynamics of, chap, xiv., 184.
Particles, system of, 194 ; four,
dynamically equivalent to rigid
body, 199.
Pascal hexagon, 121.
Pedal of quadric, 109 ; of three-system
of screws, 164.
Permanent screws, 209.
Permutation, cyclical, of quaternions
under S, 16 ; cyclical, of linear func-
tions in product, 100 ; of^ symbols in
combinatorial function, 270,
Perpendicular, on line, 36 ; on plane,
36 ; to two lines, 40 ; line, to itself,
96; on tangent plane, 109; on
generator of hyperboloid, 116; on
axis of screw, 156; in astatics, 163;
of three-system, 164 ; of cyclindroid,
166 ; in hyperspace, 303.
Perspective, 46, 278.
Perturbed orbit, 188.
Pfaff, 86.
Philosophical transactions, 101, 263, 275.
Pitch, of ruled surface, 138 ; of screw,
156; in astatics, 161; of three-system,
164; of two-system, 165; of finite
displacement, 169 ; of impulsive and
of instantaneous, 202.
Plane of quaternion, 13 ; straight line
and, chap, v., 35; polar, for sphere,
50 ; for quadric, 108 ; radical, 50 ;
inverse of, 53; cyclic, 113; osculat-
ing, 132; generating developable, 135;
of no virial, 157 ; central, in astatics,
160 ; of elastic symmetry, 245 ;
polarised wave in elastic solid, 247 ;
in dielectric, 257 ; protective symbol
for, 265 ; equation of, 266 ; united, of
linear transformation, 274 ; to qua-
dric, polar, 276.
Plticker's coordinates of a line repre-
sented by (or, r), 40.
Poinsot, central axis, 156.
Point, stationary, 63, 83 ; of inflexion,
83 ; circular, 96, 126, 307 ; double,
on wave-surface, 248, 261 ; on
Jacobiaii, 298 ; apparent, 292 ; sym-
bol, 263 et seq. } 308 ; united, of linear
transformation, 274, 276.
Polar, harmonic, 46; plane of point
with respect to sphere, 50 ; to qua-
dric, 108, 276 ; line to quadric, 109 ;
developable, 136, 139 ; general theory
of, 296.
Polarised waves in elastic solid, 247 ; in
dielectric, 258.
Pole, see Polar.
Poles, spherical harmonic referred to its,
224.
Polhode, 198.
Polygon, inscribed to sphere, 55 ; in-
scription of, 56 ; loci related to vari-
able, 300.
Potential, operator V 2 , 220; expres-
sion for, 223 ; surfaces, equi-, 227 ;
velocity, due to vortices, 235.
Power of vector, 28, 69, 173 ; of quater-
nion, 29 ; of point with respect to
sphere, 49, 50.
Poynting flux of radiated energy, 253 ;
parallel to ray-velocity, 257.
Principal, axes of section of quadric,
111; normal to curve, 134; curva-
ture, 143 ; axes of inertia, 197 ; screws,
209; circuit, 232.
Product, of two vectors denned, 8 ;
associative property of, 11 ; reciprocal
of, 12 ; of two quaternions, construe-
tion for, 14 ; conjugate of, 15 ;
318
INDEX.
spherical representation of, 16, 31 ;
differential of, 68 ; of linear functions,
95, 100; of vectors in hyperspace,
303 ; Grassmann's, 306.
Projection, of point on plane, 37 ; of
ellipse into circle, 83 ; of vectors, in-
variants relating to, 98 ; of curvature,
141.
Protective geometry, chap, xvn., 263;
in hyperspace, SOS.
Propagation of disturbance, 255.
Proportional to three vectors, fourth, 31.
Pure strain, 177 ; converting general
wave-surface into Fresnel's, 262.
Pyramid or system of three planes, ex-
amples on, 38; invariant relations
for, 98.
Quadratic equation satisfied by quater-
nion, 29 ; by special linear function,
95 ; quaternion function, 274.
Qnadric surfaces, chap, ix., 106; an-
harmonic generation of, 65 ; oscula-
ting surface, 144; pitch, 165 ; elonga-
tion, 181 ; general, in point symbols,
275 ; inscribed in developable, 279 ;
tangent-line developable for, 282 ;
invariants of, 288.
Quadrilateral, spherical, 34 ; complete,
46 ; inscribed to sphere, 56 ; common
to quadric and linear complex, 274.
Quadrimonial form, for quaternion, 13 ;
for linear quaternion function, 272.
Quartic, Steiner's, 164, 166 ; symbolic,
of linear quaternion function, 274.
Quaternion, as sum of scalar and vector,
9 ; as product of two vectors, 9 ; as
function of quaternions, 12; as quo-
tient of vectors, 13 ; as operator, 14 ;
as power of vector, 28 ; anharmonic,
56 ; invariants of linear vector func-
tion, 97, 159, 212 ; moment of force,
157; as symbol of point, 263; of
plane, 265; function, linear, 272
et seq.
Quotient, of parallel vectors, 3; of
vectors, 13,
Radical plane of spheres, 50 ; axis, 51 ;
centre, 52, 53.
Radius of quadric, 107, see Conjugate,
Curvature,
Rank of curve, 292.
Ratio, of vectors, 13 ; of torsion to cur-
vature, constant, 137.
Ray-velocity, 248, 257.
Rayleigh, Lord, 240.
Reaction, 194, 200, 236 ; of constraint,
189, 204.
Reality of roots of self-conjugate vector
function, 96; of principal screws,
209 ; of united points, 276.
Reciprocal, of vector, 11 ; of product,
12 ; of quadric, 110 ; screws, 204 ; of
quadric, 276.
Reciprocity, for a surface, relations of,
294.
Reciprocation, quadric of, 266, 309.
Rectangular vectors, system of three
mutually, 10 ; relations connecting
two systems of, 33 ; axes of function,
96, 97 ; vectors transformed from,
98 ; normals to confocals, 123 ; re-
lated to curve, 134 ; to surface, 146 ;
examples relating to, 173 ; in hyper-
space, 303.
Rectifying developable, 136, 139.
Reduced wrench, 205.
Reflection in plane mirror, 19 ; in
moving mirror, 130 ; of force for
brachistochrone, 193.
Refraction, 22 ; astronomical, 85, 175 ;
conical, 248, 260.
Regression, edge of, 136 ; aee Develop-
able.
Regular solids, rotations related to,
104.
Relative, magnitudes and directions of
two vectors, 13 ; motion, 171.
Remainder of a series, 79.
Resolution of vector into components,
chap, in., 23; of linear function,
96, 99 ; of strain, 178 ; of force, tan-
gential and normal, 185 ; of linear
quaternion function, 282.
Resultant of statical forces, 156.
Revolution, cone of, 45 ; cylinder of,
45 ; condition for quadric of, 114;
tangent cylinder of, 115; motion of
particle on, 190.
Rigid, see Body, Dynamics.
Root, of a quaternion, ?i th , 28 ; differ-
ential of square-, 77 ; of linear vec-
tor function, latent, 93 ; square-, 99 ;
linear function, symbolic, w th , of
unity, 105; linear quaternion function,
latent, of, 272, 276 ; square- of, 282.
Rotation, convention respecting sense
of, 7; conical q.v. t 18; forces, 160;
finite displacement, 168 et seq. ;
strain, 177 et >>eq., 182 ; of elements,
212 ; in hyperspace, 307.
Roulette, 83, 84.
Royal Irish Academy, see Academy.
Ruled, hyperboloid q.v., 65, 116, 264,
270 ; surfaces, 128, 137 et seq. ; sur-
face, differential equation of, 149.
153.
Russell, Robert, 22, 61.
S symbol for scalar, 6, 19 ; differential
of 80, 68.
Salmon, 114, see Geometry of Three
Dimensions, Higher Plane Curves.
INDEX
319-
Scalar, 3, 6 ; of product, 15 ; point,
263.
Screws, theory of, applied to, motion
of emanant, 137 ; statics, 156, 159,
163 ;, displacements, 169; dynamics,
200, 204
Segments, theorem of six, 46.
Self-conjugate, tetrahedron to sphere,
52, 276 ; vector function, 80, 96, 97 ;
tetrahedron of two quadric?, 277.
Sense of rotation, 7.
Series, exponential, 28 ; Taylor's 79 ;
of spherical harmonics, 223, 224.
Sextic curve, Jacobian, 293, 295.
Shaw, J. B., 263.
Shear, 178.
Shortest distance between lines, 40,
138, 154.
Similitude, direct and inverse, 14.
Six, coordinates of line (<r, T), 40 ; seg-
ments, 46 ; constants of self-con-
jugate function, 96 ; screws, 166 ;
co-reciprocal, 206.
Sixteen, constants in linear quater-
nion function, 272 ; square roots of
linear quaternion function, 282.
Solenoidal distribution of vectors, 234.
Solid, harmonic, 70, 76, 222 et scq. ;
elastic, 222, 239, 242 et seq. ; mov-
ing in fluid, 241.
Solution of equations, involving linear
function, 92, 117 ; involving V, 218.
Sphere, chap, vi., 49 ; inversion of,
52 ; through four points, 53, 55, 58 ;
touching four planes, 54 ; and poly-
gon, 55, 56 ; solid, 59 ; generating
ellipsoid, 115 ; osculating, 136 ; en-
velope of, 151 ; surface generated by,
155 ; of reciprocation, unit, 266.
Spherical, trigonometry, chap. iv. , 29 ;
excess, 33 ; harmonics, 70, 76, 222 ;
curvature, 136 ; astronomy, exam-
' pies, 174.
Sphero-conic, 118.
Spin-vector, 96, 97 ; of ^ 97 ; in strain,
181, 182; of element, 212.
Spiral, logarithmic, 82.
Square-root of quaternion, differential
of, 77; of linear function, 99, 112,
124, 177 ; of linear quaternion func-
tion, 282.
Standard form of V, 75 ; of two linear
functions, 100 ; of screw-system, 164 ;
of quadric and linear complex, 278.
Statics, chap. XL, 156.
Steiner's quartic, 164, 166.
Stokes's theorem, 215.
Storage of energy, elastic solid, 243 ;
electric and magnetic, 252.
Strain, chap, xm., 177, 212,238; stress
in terms of, 243.
Strength of tube, 233, 235.
Stress, 237 et seq. ; in viscous fluid, 238 ;
in iso tropic solid, 239 ; in terms of
strain, 243 ; in electro-magnetic field,
255.
Striction, line of, 138 ; of quadric, 140.
Subtraction of vector, 2.
Sum of vectors, 2 ; of scalar and vector,
9; of quaternions, 9; of weighted
points, 264 ; of area vectors in hyper-
space, 306.
Supplemental triangles, 29 ; related to
axes of function and conjugate, 94 ;
to propagation of light, 258.
Surface, in terms of parameters, 64 ;-
quadric, chap, ix., 106; non-central^.
117; of centres, 125; ruled, 137;
curvature of, 141 ; generated by circle,
154 ; equilibrium of chain on, 167 ;
motion of particle on, 189 ; of dis-
continuity, 216 ; wave-, 248, 261 ; of
centres, generalized, 287 ; general,
293.
Surfaces, families of, 148; equipotential,
227 ; characteristic, in optics, 228.
Symbol, 19, V, 70, 211 ; are Operator;
(fj., A) for screw, 163 ; zfr and g de-
fined, 229 ; point-, 263, 308. ~
Symbolic, multiplication table, S, V, K,
T, U, 19 ; vector, V, 75 ; form of
Taylor's theorem, 79 ; cubic of linear
function, 93, 100 ; case of depressed,
95 ; quartic of linear quaternion
function, 274.
Symmetry, elastic, 245.
T symbol for tensor, 4, 12, 19 ; differ-
ential of Tg, 68 ; development of
T(p + 0), 85.
Tait, P. G., referred to, 7, 20, 33, 99,
163, 192, 211, 214, 218.
Tangent, to sphere, 49 ; curve, 63 ;
surface, 65 ; quadric, 108 ; confocal,
124 ; generalized confocals, 280 ; line
developable of two quadrics, 282;
conjugate, 295 ; meeting surface in
four consecutive points", 298.
Tangential equation of quadric, 110 ;
and normal components of force, 185,
189 ; transformation, 273 ; equation
of quadric and linear complex, 276.
Taylor's series, 79.
Telescope, examples on composition of
rotations, 175.
Tensor of vector, 4; quaternion, 12;
biquaternion, 20; of sum, develop-
ment of, 85.
Tetrahedra, in perspective, 46 ; corre-
sponding vertices of, joined by gen-
erators of hyperboloid, 103 ; recip-
rocal, of united points of linear
transformation and its conjugate*,
274.
INDEX.
Tetrahedral, coordinates, q.v., 268 ;
complex, 302.
Tetrahedron, formulae relating to, 42 ;
regular, 45, 104; anharmonic rela-
tions of point and, 46 ; self-conjugate
to sphere, 52 ; and sphere, 53 ; forces
on edges of, 158 ; of reference, 268 ;
self -conjugate to two quadrics, 277 ;
invariants relating to, 288 ; loci re-
lating to variable, 300.
Thomson and Tait, 200.
Three-bar motion, 60, 85; -system of
screws, 164, 205; plane polarised
waves in solid, 248.
Tore, 59.
Torsal generator, 140.
Torse, see Developable,
Torsion, 132, 134.
Total curvature, 148.
Transformation, effected by linear vec-
tor function, 89 ; by self -con jugate
function, 97; Cremona, 101; of screws,
208 ; general linear, 272, et seq. ; ex-
amples of, 285 ; invariants of, 288.
Transversals of lines, 103.
Triangle, and point, harmonic proper-
ties, 45 ; and conic, 48.
Trigonometry, formulae for plane, 25 ;
de Moivre's theorem, 27 ; spherical,
29.
'Trilinear function, 243.
Trinomial form for linear function, 89 ;
for pair of functions, 100.
Tube, motion of particle in rotating,
191 ; in fluid motion, 233, 235.
Twist of curve, vector, 133 ; -velocity,
170, 171, 201.
Twisted, cubic q.v., 93, 104, 109, 123,
133, 172, 267, 278, 285, 293.
Two linear functions, 100 ; reduction
to, forces, 158 ; angular velocities,
172.
U, symbol for \ersor, 4, 13, 19 ; differ-
erential of Ug, 68 ; development of
U(p + q], 85.
Umbilical generator, 125 ; generalized,
279, 286.
Unicursal curve, 152.
.Unit, of length, 4 ; vector denoted by
Ua, 4 ; vectors, system of mutually
(q.v.) rectangular, 10, 96, 98, 134, 146;
point of anharmonic coordinates, 44 ;
weight, points of, 263 ; sphere of re-
ciprocation, 266; vectors in hyper-
space, 303.
United, screws, 209 ; points of trans-
formation, 274, 276.
V, symbol for vector, 7, 19; differ-
ential of Vq, 68.
Variable origin, 157.
Variations, calculus of, 192 ; of inte-
grals, 231, 233.
Vector, as directed right line, 1 ; as
operator, 14 ; arc, 17 ; area, 23 ;
of curve, 62 ; of surface, 64 ; func-
tion, linear, 88 ; spin-, 96 ; equation
of confocals, 124 ; emanant, 131 ; as
difference of two points, 263; as
point at infinity, 263 ; area, in
hyperspace, 303.
See Linear vector function, etc.
Vectorial algebra, 11.
Velocities, virtual, 157, 254.
Velocity, 63 ; hodograph, 83, 187, 189 ;
of emanant, 138 ; twist-, 170, 201 ;
relative, 171 ; of particle, 184 etseq. ;
areal, 186, 188 ; of element of
medium, angular, 212; potential,
due to vortex rings, 235 ; wave-,
247, 257 ; ray-, 248, 257.
Versor of vector, 4, 14 ; of quaternion,
13, 16.
Versors, coplanar, chap, iv,, 27.
Vibration of particle, 185.'
Virial, 157.
Virtual, velocities, 157, 254; co-effi-
cient of two screws, 206.
Viscous fluid, 238, 240.
Volume of parallelepiped, 23 ; of tetra-
hedron, 265, 269 ; directed, 304.
Vortex motion, 235, 238.
Wave-surface, 163, 248, 261 ; velocity.
247, 257.
Waves, propagation of disturbance by,
255.
Weierstrass, 199.
Wrench, in statics, 156, 163 ; im-
pulsive, 201, 204; evoked, 204;
reduced, 205.
Zero, square-root of, 29.
GLASGOW: PRINTED
AT THE UNIVERSIT7 PRESS BY ROBERT MACLEH08E AND CO. LTD.
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11